6  ELLIPTIC  FUNCTIONS. 

i\^  and  P  being  rational  integral  functions  of  x.     Whence  Eq. 

(3)  becomes 

(4)  F=    [ Ndx-    i^"^"" 


R 


Eq.  (4)  shows  that  the  most  general  form  of  Fcan  be  made 
to  depend  upon  the  expressions 

(5)  ^'=/^. 

and 

fNdx. 

This  last  form  is  rational,  and  needs  no  discussion  here. 
We  can  write 

G,  +  G^x^  +  G,x^  4-  .  .  .  +  (g^  -f-  g^;^«  -I-  .  .  .)^ 
-  H,^H,x'^^H,x^-\-.  .  .  ^(H,-{-H,x^-]-..  .)x' 

Multiplying  both  numerator  and  denominator  by 

H,  +  H,x^-\-H,x'-\-.  .  .-{H,  +  H^x'  +  H,x*-\-.  .  .)x, 

we  have  a  new  awaerator  which  contains  only  powers  of  x''. 
The  result  takes  the  following  form  : 

N.-^N.x-'  +  N.x'-^N.x'-^... 
=  0{x')  +  W{x') .  X. 

Equation  (5)  thus  becomes 


(6)         F'  =  f^(^y^  +  fY(^')  '^'^^ 


R 


ELLIPTIC  INTEGRALS.  7 

We  shall  see  presently  that  R  can  always  be  assumed  to  be 
of  the  form 


Therefore,  putting  x'  =  z,  the  second  integral  in  Eq.  (6) 
takes  the  form 


k'zY 


which  can  be  integrated  by  the  well-known  methods  of  Integral 
Calculus,  resulting  in  logarithmic  and  circular  transcendentals. 
There  remains,  therefore,  only  the  form 


/ 


^{je)dx 


R 
to  be  determined. 

We  will  now  show  that  R  can  always  be  assumed  to  be  in 
the  form 


V{l  -x'){i  -  yfeV). 
We  have 


R  =  V  Ax*-{-  Bx'  -]r  Cx'  ^  Dx  -\- E 
=  VG{x  -  a\x  -  b){x  -  c){x  -  d\ 

a,  b,  c,  and  d  being  the  roots  of  the  polynomial  of  the  fourth 
degree,  and  G  any  number,  real  or  imaginary,  depending  upon 
the  coefficients  in  the  given  polynomial. 
Substituting  in  equation  (i) 

^    P±jy 

we  have 

(7)  V=fcj>{y,p)dy, 


THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

LOS  ANGELES 


Digitized  bythe  Internet  Archive 

in  2007  with  funding  from 

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http://www.archive.org/details/ellipticfunctionOObakeiala 


Elliptic  Functions. 


An  Elementary  Text-Book  for 
Students  of  Mathematics. 


BY 

ARTHUR  L.   BAKER,  C.E.,  Ph.D., 

Professor   of    Mathematics    in   the    Stevens    School  of   the   Stevens    Institute  of 
Technology,   Hoboken.  N.  J.  ;   formerly   Professor   in  the   Pardee 
Scientific  Department,  Lafayette  College,  Easton,  Pa. 


r        H{ti) 

sin  am  u  =  ——  •  y-. — -. 


NEW   YORK: 

JOHN   WILEY   &   SONS, 

53  East  Tenth  Street. 


Copyright,  1890, 

BY 

Arthur  L.  Baker. 


Robert  DRtrnMOND,  Ferris  Bros., 

Electrotyper,  Printers, 

444  &  446  Pearl  Street,  326  Pearl  Street, 

New  York.  *<ew  York. 


Libraiy 


3H-3 

311^ 


PREFACE. 


In  the  works  of  Abel,  Euler,  Jacobi,  Legendre,  and  others, 
the  student  of  Mathematics  has  a  most  abundant  supply  of 
material  for  the  study  of  the  subject  of  Elliptic  Functions. 

These  works,  however,  are  not  accessible  to  the  general 
student,  and,  in  addition  to  being  very  technical  in  their  treat- 
ment of  the  subject,  are  moreover  in  a  foreign  language. 

It  is  in  the  hope  of  smoothing  the  road  to  this  interesting 
and  increasingly  important  branch  of  Mathematics,  and  of 
putting  within  reach  of  the  English  student  a  tolerably  com- 
plete outline  of  the  subject,  clothed  in  simple  mathematical 
language  and  methods,  that  the  present  work  has  been  com- 
piled. 

New  or  original  methods  of  treatment  are  not  to  be  looked 
for.  The  most  that  can  be  expected  will  be  the  simplifying  of 
methods  and  the  reduction  of  them  to  such  as  will  be  intelligi- 
ble to  the  average  student  of. Higher  Mathematics. 

I  have  endeavored  throughout  to  use  only  such  methods  as 
are  familiar  to  the  ordinary  student  of  Calculus,  avoiding  those 
methods  of  discussion  dependent  upon  the  properties  of  double 
periodicity,  and  also  those  depending  upon  Functions  of  Com- 
plex Variables.  For  the  same  reason  I  have  not  carried  the 
discussion  of  the  0  and  H  functions  further. 


IV  PREFACE. 

Among  the  minor  helps  to  simplicity  is  the  use  of  zero 
subscripts  to  indicate  decreasing  series  in  the  Landen  Trans- 
formation, and  of  numerical  subscripts  to  indicate  increasing 
series,  I  have  adopted  the  notation  of  Gudermann,  as  being 
more  simple  than  that  of  Jacobi. 

I  have  made  free  use  of  the  following  works :  Jacobi's 
Fundamenta  Nova  Theoriae  Func.  Ellip.;  Houel's  Calcul 
Infinitesimal ;  Legendre's  Traits  des  Fonctions  Elliptiques  ; 
Durege's  Theorie  der  Elliptischen  Functionen  ;  Hermite's 
Theorie  des  Fonctions  Elliptiques;  Verhulst's  Theorie  des 
Functions  Elliptiques ;  Bertrand's  Calcul  Integral :  Lau^ 
rent's  Theorie  des  Fonctions  Elliptiques  ;  Cayley's  Elliptic 
Functions  ;  Byerly's  Integral  Calculus  ;  Schlomilch's  Die 
Hoheren  Analysis;  Briot  et  Bouquet's  Fonctions  Ellip- 
tiques. 

I  have  refrained  from  any  reference  to  the  Gudermann  or 
Weierstrass  functions  as  not  within  the  scope  of  this  work, 
though  the  Gudermannians  might  have  been  interesting 
examples  of  verification  formulae.  The  arithmetico-geometrical 
mean,  the  march  of  the  functions,  and  other  interesting  investi- 
gations have  been  left  out  for  want  of  room. 


CONTENTS. 


Introductory  Chapter,    . 
Chap.  I.  Elliptic  Integrals, 
II.  Elliptic  Functions, 

III.  Periodicity  of  the  Functions, 

IV.  Landen's  Transformation, 
V.  Complete  Functions,     . 

VI.  Evaluation  for  0, 
VII.  Factorization  of  Elliptic  Functions, 
VIII.  The  0  Function, 
IX.  The  &  AND  H  Functions, 
X.  Elliptic  Integrals  of  the  Second  Order, 
XI.  Elliptic  Integrals  of  the  Third  Order, 
Xll.  Numerical  Calculations.  ^,    . 

XIII.  Numerical  Calculations,  A',  . 

XIV.  Numerical  Calculations,  u,    . 
XV.  Numerical  Calculations,  0,    . 

XVI.  Numerical  Calculations,  £{^,  (p), 
XVII.  Applications, 


PAGE 
I 

4 

i6 

22 
30 

45 

48 
51 

66 

69 

81 

go 

94 

98 

102 

108 

III 

"5 


/ 


ELLIPTIC   FUNCTIONS. 


INTRODUCTORY  CHAPTER* 

The  first  step  taken  in  the  theory  of  Elliptic  Functions 
was  the  determination  of  a  relation  between  the  amplitudes  of 
three  functions  of  either  order,  such  that  there  should  exist  an 
algebraic  relation  between  the  three  functions  themselves  of 
which  these  were  the  amplitudes.  It  is  one  of  the  most  re- 
markable discoveries  which  science  owes  to  Euler.  In  1761  he 
gave  to  the  world  the  complete  integration  of  an  equation  of 
two  terms,  each  an  elliptic  function  of  the  first  or  second  order, 
not  separately  integrable. 

This  integration  introduced  an  arbitrary  constant  in  the 
form  of  a  third  function,  related  to  the  first  two  by  a  given 
equation  between  the  amplitudes  of  the  three. 

In  1775  Landen,  an  English  mathematician,  published  his 
celebrated  theorem  showing  that  any  arc  of  a  hyperbola  may 
be  measured  by  two  arcs  of  an  ellipse,  an  important  element 
of  the  theory  of  Elliptic  Functions,  but  then  an  isolated  result. 
The  great  problem  of  comparison  of  Elliptic  Functions  of  dif- 
ferent moduli  remained  unsolved,  though  Euler,  in  a  measure, 
exhausted  the  comparison  of  functions  of  the  same  modulus. 
It  was  completed  in  1784  by  Lagrange,  and  for  the  computation 

*  Condensed  from  an  article  by  Rev.  Henry  Moseley,  M.A.,  F.R.S. ,  Prof,  of 
Nat.  Phil,  and  Ast.,  King's  College,  London. 


2  ELLIPTIC  FUNCTIONS. 

of  numerical  results  leaves  little  to  be  desired.  The  value  of 
a  function  may  be  determined  by  it,  in  terms  of  increasing  or 
diminishing  moduli,  until  at  length  it  depends  upon  a  function 
having  a  modulus  of  zero,  or  unity. 

For  all  practical  purposes  this  was  sufficient.  The  enor- 
mous task  of  calculating  tables  was  undertaken  by  Legendre. 
His  labors  did  not  end  here,  however.  There  is  none  of  the 
discov.eries  of  his  predecessors  which  has  not  received  some 
perfection  at  his  hands ;  and  it  was  he  who  first  supplied  to 
the  whole  that  connection  and  arrangement  which  have  made 
it  an  independent  science. 

The  theory  of  Elliptic  Integrals  remained  at  a  standstill 
from  1786,  the  year  when  Legendre  took  it  up,  until  the  year 
1827,  when  the  second  volume  of  his  Traits  des  Fonctions 
Elliptiques  appeared.  Scarcely  so,  however,  when  there  ap- 
peared the  researches  of  Jacobi,  a  Professor  of  Mathematics  in 
Konigsberg,  in  the  123d  number  of  the  Journal  of  Schumacher, 
and  those  of  Abel,  Professor  of  Mathematics  at  Christiania,  in 
the  3d  number  of  Crelle's  Journal  for  1827. 

These  publications  put  the  theory  of  Elliptic  Functions 
upon  an  entirely  new  basis.  The  researches  of  Jacobi  have  for 
their  principal  object  the  development  of  that  general  relation 
of  functions  of  the  first  order  having  different  moduli,  of  which 
the  scales  of  Legrange  and  Legendre  are  particular  cases. 

It  was  to  Abel  that  the  idea  first  occurred  of  treating  the 
Elliptic  Integral  as  a  function  of  its  amplitude.  Proceeding 
from  this  new  point  of  view,  he  embraced  in  his  speculations 
all  the  principal  results  of  Jacobi.  Having  undertaken  to  de- 
velop the  principle  upon  which  rests  the  fundamental  proposi- 
tion of  Euler  establishing  an  algebraic  relation  between  three 
functions  which  have  the  same  moduli,  dependent  upon  a  cer- 
tain relation  of  their  amplitudes,  he  has  extended  it  from  three 
to  an  indefinite  number  of  functions;  and  from  Elliptic  Func- 
tions to  an  infinite  number  of  other  functions  embraced  under 
an  indefinite  number  of  classes,  of  which  that  of  Elliptic  Func- 


INTRODUCTORY  CHAPTER.  3 

tions  is  but  one  ;  and  each  class  having  a  division  analogous  to 
that  of  Elliptic  Functions  into  three  orders  having  common 
properties. 

The  discovery  of  Abel  is  of  infinite  moment  as  presenting 
the  first  step  of  approach  towards  a  more  complete  theory  of 
the  infinite  class  of  ultra  elliptic  functions,  destined  probably 
ere  long  to  constitute  one  of  the  most  important  of  the 
branches  of  transcendental  analysis,  and  to  include  among  the 
integrals  of  which  it  effects  the  solution  some  of  those  which  at 
present  arrest  the  researches  of  the  philosopher  in  the  very 
elements  of  physics. 


CHAPTER   I. 

ELLIPTIC   INTEGRALS. 
The  integration  of  irrational  expressions  of  the  form 


Xdx  VA  -\-  Bx-[-  Cx\ 
or 

Xdx 


VA  -\-Bx-{-  Cx' ' 

X  being  a  rational  function  of  x,  is  fully  illustrated  in  most  ele- 
mentary works  on  Integral  Calculus,  and  shown  to  depend  upon 
the  transcendentals  known  as  logarithms  and  circular  functions, 
which  can  be  calculated  by  the  proper  logarithmic  and  trigono- 
metric tables. 

When,  however,  we  undertake  to  integrate  irrational  expres- 
sions containing  higher  powers  of  x  than  the  square,  we  meet 
with  insurmountable  difificulties.  This  arises  from  the  fact  that 
the  integral  sought  depends  upon  a  new  set  of  transcendentals, 
to  which  has  been  given  the  name  of  elliptic  functions,  and 
whose  characteristics  we  will  learn  hereafter. 

The  name  of  Elliptic  Integrals  has  been  given  to  the  simple 
integral  forms  to  which  can  be  reduced  all  integrals  of  the  form 

(I)  V=fF{X,R)dx, 

where  F{X,  R)  designates  a  rational  function  of  x  and  R,  and 
R  represents  a  radical  of  the  form 


R  =  VAx'  +  Bx'  +  Cx'  -^Dx-\-E, 


ELLIPTIC  INTEGRALS.  5 

where  A,  B,  C,  D,  E  indicate  constant  coefficients. 

We  will  show  presently  that  all  cases  of   Eq.  (i)   can  be 
reduced  to  the  three  typical  forms 


i 


dx 


x^dx 


(2) 


-  x'){\  -/&V)' 
dx 


{x'  +  a)  |/(i  —  x'){i  -  k'x')  ' 


which  are  called  elliptic  integrals  of  the  first,  second,  and  third 
order. 

Why  they  are  called  Elliptic  Integrals  we  will  learn  further 
on.  The  transcendental  functions  which  depend  upon  these 
integrals,  and  which  will  be  discussed  in  Chapter  IV,  are  called 
Elliptic  Ficnctions. 

The  most  general  form  of  Eq.  (i)  is 


(3)  ^=/f^^^^-^-. 


A-\-BR 
C  +  DR' 


where  A,  B,  C,  and  D  stand  for  rational  integral  functions  of  x. 

A  +  BR         ^        . 
7=r— j — tyd  can  be  written 
L  -j-  L/K 

A  +  BR  _AC-  BDR'  _  (AD  -  CB)R'    £ 
C-^DR~    C'-D'R'  ~     C^-D'R'     '  R 


6  ELLIPTIC  FUNCTIONS. 

N  and  P  being  rational  integral  functions  of  x.     Whence  Eq. 

(3)  becomes 

(4)  V=JNdx-j?^. 

Eq.  (4)  shows  that  the  most  general  form  of  Fcan  be  made 
to  depend  upon  the  expressions 

(5)  ^'=/^. 

and 

jNdx. 

This  last  form  is  rational,  and  needs  no  discussion  here. 
We  can  write 


H,^H,x^H,x'-\-.  .  . 

G,  +  G,x^+G,x*+.  .  .  +  (g.  +  g.^'  +  -  •  0^ 
-I/,  +  II,x^  +  H,x'+  .  .  .  -^{H,-\-N,x'-\-..  .)x' 

Multiplying  both  numerator  and  denominator  by 

H,  +  H^x^-\-H,x^-\-.  .  .  -  {H,-\-H,x^JrH.^'+-  •  •)^, 

we  have  a  new  ftwaerator  which  contains  only  powers  of  x^. 
The  result  takes  the  following  form : 

p  _  M,  +  M^x-^  J^M,x^-\-...+  {M,  +  M,x^  +  M,x'  +  ■  •  •>• 
N.  +  N.x-'  +  N.x'  +  N.x'-^-... 

=  ^{x"")  +  W{x-')  .X. 
Equation  (5)  thus  becomes 


(6)  v  =  fii^)^  +  fY(^')  '^-^^ 


R 


ELLIPTIC  INTEGRALS.  7 

We  shall  see  presently  that  R  can  always  be  assumed  to  be 
of  the  form 


V{i  -  x'){i  -  k'x'). 

Therefore,  putting  x"  =  z,  the  second  integral  in  Eq.  (6) 
takes  the  form 


-k'z) 

which  can  be  integrated  by  the  well-known  methods  of  Integral 
Calculus,  resulting  in  logarithmic  and  circular  transcendentals. 
There  remains,  therefore,  only  the  form 


/ 


^{x')dx 


R 
to  be  determined. 

We  will  now  show  that  R  can  always  be  assumed  to  be  in 
the  form 


|/(i  -x'){i  -/feV). 
We  have 


R=V  Ax*-{-  Bx'  -]-  Cx'  ^  Dx  -[- E 
=  VG{x  -  dix  -  b){x  -  c){x  -  d\ 

a,  b,  c,  and  d  being  the  roots  of  the  polynomial  of  the  fourth 
degree,  and  G  any  number,  real  or  imaginary,  depending  upon 
the  coeflficients  in  the  given  polynomial. 
Substituting  in  equation  (i) 

p-\-gy 
^  =  — i — > 

we  have 

(7)  V=f(P{y,p)dy, 


8  ELLIPTIC  FUNCTIONS. 

p  designating  the  radical 

P=  ^G[p-a-\-{q-a)y']  [p-b-\-{g- b)y-]  \^p-c^{q-c)y\  . . . 

In  order  that  the  odd  powers  of  y  under  the  radical  may 
disappear  we  must  have  their  coefficients  equal  to  zero;  i.e., 


whence 


and 


{p  -  a)  {q  -  b)  -^{p  -  b)  {q  -  d)r=  O, 
{P  -  c)\q-  d)-\-{j>  -  d){q  -  .c)  =  0  ', 

2pq  -{p-\-  q)(a  +  ^)  +  2ab  =  O, 
2pq  —  (/  +  q){c  -f  d)  -Y  2cd  =  O, 


(8) 


pq 


abic  -\-  d)  —  cd{a  -\-  b) 
a-\-b  -{c-^d)  ' 
2ab  —  2cd 

a  +  b  —  {c-\-dy 


Equation  (8)  shows  that  /  and  q  are  real  quantities,  whether 
the  roots  a,  b,  c,  and  d  are  real  or  imaginary  ;  a,  b,  and  c,  d  being 
the  conjugate  pairs. 

Hence  equation  (i)  can  always  be  reduced  to  the  form  of 
equation  (7),  which  contains  only  the  second  and  fourth  powers 
of  the  variable. 

This  transformation  seems  to  fail  when  a-\-b  —  {c-\-d)  ^o\ 
but  in  that  case  we  have 

R=  VGlx'  -  (a  +  b)x  +  ab-]  [x'  -  («  +  b)x  +  cd\ 

and  substituting 

a  +  b 
X  ^  y 

will  cause  the  odd  powers  of  y  to  disappear  as  before. 
If  the  radical  should  have  the  form 


VG{x  —  d){x  —  b){x  —  c), 


ELLIPTIC  INTEGRALS, 
placing  X  ^  y^  -\-  a,  we  get 

^—  J  ^{y^  pyyy 


<p  designating  a  rational  function  of  j  and  p. 

Thus  all  integrals  of  the  form  contained  in  equation  (i),  in 
which  R  stands  for  a  quadratic  surd  of  the  third  or  fourth 
degree,  can  be  reduced  to  the  form 

<9)  V=fcp{x,R)dx,  . 

j^  designating  a  radical  of  the  form 

VG{i  +  mx'){i  +  nx'), 


^m  and  n  designating  constants. 
"  It  is  evident  that  if  we  put 


x'  ^  X  V  —  m,     k^  := , 

m 


we  can  reduce  the  radTcaTToTnieTorm* 


^ 


i/(i  -  ;r')(l  -  /&V). 

We  shall  see  later  on  that  the  quantity  k^,  to  which  has 
been  given  the  name  modulus,  can  always  be  considered  real 
and  less  than  unity. 

Combining  these  results  with  equation  (6),  we  see  that  the 
integration  of  equation  (i)  depends  finally  upon  the  integration 
of  the  expression 


J     |/(i-;r')(i-/^V)      J 


(p{x^)dx 
R~' 


(6 


lO  ELLIPTIC  FUNCTIONS. 

The  most  general  form  of  0(^')  is 

=  P,  +  P.x'  +  P,x^+P,x^+.., 
Hence 

But    /  ^-^-  depends  upon    /  -^  and     /  — ^,  which   can 

be  shown  as  follows : 

Differentiating  Rx^""^,  we  have 

d[x^'"-^R]  =  dlx''"-^  Va  +  ^x'  4-  rx'] 

=  {2m  —  s)^'""'^^^  V  a-{- J3x' -\- yx' 
x^'"-^{/3x  +  2yx')dx 
'^     i/a-\-Px'  +  rx*  ' 

Integrating  and  collecting,  we  get 

j^^2m-i  _  (2w  —  3)a  /  — ^ h  (2;;^  —2)/3  I 


R 

'x''"'dx 


-\-{2m-i)y  /  — ^ 

(12)  =a'J   —^-  +  ^J    -^-J,y  j    -^ 


ELLIPTIC  INTEGRALS.  II 

Whence  we  get,  by  taking  m  =  2, 

/dx  Cx'^dx   ,         Cx'dx 

/x^dx 
o     can  be  found 

by  successive  calculations,  when  we  are  able  to  integrate  the 
expressions 

/dx  .        Cx^dx 

-R     -"    J  -W' 

the  first  and  second  of  equation  (2). 

We  will  now  consider  the  second  class  of  terms  in  eq.  (i  i)» 
Ldx 
^^^"  (P  +  aYR' 

This  second  term  is  as  follows : 

J  (^"  +  ^)"^  ~J  (^'  +  ^T^    J  C-^'  +  ^)""'^ 


Each  of  these  terms  can  be  shown  to  depend  ultimately 
upon  terms  of  the  form 

x'^dx      dx  ^  dx 

r,    and 


R   '      R'    """     {x'  +  a)R' 

The  two  former  will  be  recognized  as  the  two  ultimate  forms 
already  discussed,  the  first  and  second  of  equation  (2).  The 
third  is  the  third  one  of  equation  (2). 

This  dependence  of  equation  (14)  can  be  shown  as  follows: 


12 


ELLIPTIC  FUNCTIONS. 


We  liave 
-      xR - 


_  {x^^aY-\xdR-{-Rdx) - 2x''R{n-^i){x^^ay-'dx 
~  {x'-^ay"-' 

{x'  +  a){xdR  +  Rdx)  —  2x^R{n  -  i)dx 


{x'-]raY 


Substituting  the  value  of 


R=Voc-{-ftx''-\-  yx*     and     dR  =  {/3x  +  2yx')  -^, 


we  get 


^L(x'4-aY-U  ~ 


.{x^  +  ay 
{x'-\-d){^x'-{-2yx*+a+/3x'-{-yx')-2x\n-l){a-^/3x'-^yx*)    dx 


3r 

2{n  —  i)y 


{x'  +  ay 

+  3^r 

2(«  -  i)/3 


R 


■      +2^/? 
2(«  —  i)a 


x"  -\-aa 


{x'  +  ay 


{2n  —  5)r;i^*  —  (2«  —  4)/? 

+  3«r 


;i:*  —  {2n  —  3)fl' 
—  2a^ 


x'  -\-  aa 


{x^-\-dy 

or,  by  substituting  in  the  numerator  x'^  ^  z  —  a, 


dx 
"R 


dx 
7?' 


'-{2n-s)y2'+{2n-s)iay 
—  {2n  —  4)/3 

+  3«r 


^'-(2«-5)3«V 

4-  (2«  —  4)2a^ 

—  6«''7 

—  (2;?  —  3)0' 

-\-2a^ 


{x'  +  dy 


z-\-{2n-S)ay 
—  {2n—4)a^^ 

+  3«V 
-|-(2«  —  3)rt:ar 
-2^^yS 

-\-aa  dx 

"R  ' 


ELLIPTIC  INTEGRALS.  1 3 

or,  after  resubstituting  z  ^=.  x^  -^  a,  and  integrating, 

xR  r        dx 

-  (2«  -  4)(/5  -  ^ay)J-^^^;^ 

-  (2«  -  3)(3«=;.  -  2«^  +  a)    f.  ^-^ 


+  (2«  —  2){a'y  —  a' ft  -\-  aa) 


{x'^^ay-'R 
dx 


/dx „     r         dx r         dx 
{x'-\-dy^-^R~^'^'J   {x'-\-aY-'R'^'^'J  {x'-^d)''-\ 


,    ,    'R 

-^^J  {x'  +  aYR- 
Making  «  =  2,  we  have 

dx 

{x'  +  a)R 


xR  r(x*-\-d)dx  fdx  r 


J   {^"  +  dfR' 


Equation  (i6)  shows  that 

dx 


/< 


{x^  +  «)'i? 

depends  upon  the  three  forms 

/x'^dx  Cdx  r      dx 


14  ELLIPTIC  FUNCTIONS. 

the  three  types  of  equation  (2),  and  equation  (15)  shows  that 
the  general  form 

dx 


/. 


depends  ultimately  upon  the  same  three  types. 

We  have  now  discussed  every  form  which  the  general  equa- 
tion (i)  can  assume,  and  shown  that  they  all  depend  ultimately 
upon  one  or  more  of  the  three  types  contained  in  equation  (2). 

These  three  types  are  called  the  three  Elliptic  Integrals  of 
the  first,  second,  and  third  kind,  respectively. 

Legendre  puts  x  =  sin  (p,  and  reduces  the  three  integrals 
to  the  following  forms : 

(,7)     Fik,4>)=    f*  ''1' 


Vi  -  k'  sin'  0 


^i     Tr^l^ii^-Fj/^      Vi-^'sin'0.^0; 
(18)        n{n,  k,<S>)=    ^  "^"^ 


{i  —  n  sin^  0)  1/1  —  /^'  sin'  0' 
the  first  being  Legendre's  integral  of  the  first  kind ;  the  form 

(19)  ^(^'  ^)=   I      Vi  -  k'  sin'  0  .  ^0 

being  the  integral  of  the  second  kind ;  and  the  third  one  being 
the  integral  of  the  third  kind. 

The  form  of  the  integral  of  the  second  kind  shows  why  they 
are  called  Elliptic  Integrals,  the  arc  of  an  elliptic  quadrant 
being  equal  to 


a\     V I  —  e""  sirv'  (p  .  d(f), 
*J  0 

0  being  the  complement  of  the  eccentric  angle. 


ELLIPTIC  INTEGRALS.  15 

By  easy  substitutions,  we  get  from  Eqs.  (17),  (18),  and  (19) 
the  following  solutions : 


r*sm'<t>^         F-E 


cos'  0  ^         E  -{i  -  k^)F 


'*tan'0,,       At3.n4>-E 

___  ^0  =  — TZT^— ; 


sec"  0  ,,       J  tan  0  +  (I  —  /&')F  —  E 

j-^<^  = rm^ 


/ 

/*  I                   ^     ( T-       ^^  sin  0  cos  0\ 
^  ^0  =  --3^1^^ J j; 

/*sin''0  ,^            I     (E  —  {i—k^)F      sin  0  cos  0\ 
-^^0  =  -_^(^ j^ — J — ^j; 

/''"cos"  0  ,^       F—E,   sin  0  cos  0 

A 


^' 


"t- 


Tf 


t-  -^x 


2. 


-/^ 


H 


Let  u  = 


CHAPTER  II. 
ELLIPTIC   FUNCTIONS. 


\   V\  -k'  sm'  (p 


0^  is  called  the  amplitude  corresponding  to  the  argument  «, 
and  is  written 

0  =  am  {u,  k)  =  am  u. 
The  quantity  k  is  called  the  modulus,  and  the  expression 


Vi  —  k^  sin^  0  is  written  * 


\/\  —  a  sin'  0  =  z/  am  ?/  =  z/0, 

and  is  called  the  delta  function  of  the  amplitude  o*  u,  or  delta 
of  0,  or  simply  delta  <p. 

u  can  be  written 

u  =  F{k,  0). 

The  following  abbreviations  are  used : 

_  sin  0  =:  sin  am  z^  =  sn  f  «  ; 
cos  0  ^  cos  am  ?^  ^  en  f  «  ; 

A(j)  =  J  am  //  =  dn  f  ?/  ==  Au  ; 
tan  0  =  tan  am  u  =  tn  u. 

Let  0  and  ^  be  any  two  arbitrary  angles,  and  put 

0  =  am  u  ; 
^  =  am  V. 

*  Legendre. 

\  Gudermann ,  \n  his  "  Theorie  der  Modularfunctionen  "  :   Crelie's  Journal, 
Bd.  1 8. 

i6 


ELLIPTIC  FUNCTIONS.  ly 


In  the  spherical  triangle  ABC  we  have  from 
Trigonometry,  c  and  C  being  constant, 


d<f>     ,      dib 


B 


COS  B  "^  cos  A 

Since  C  and  c  are  constant,  denoting  by  k  an  arbitrary  con- 
stant, we  have 

sin  6"       , 

(i)  -. =  k. 

^  '  sm  /i 

But 

sin  B        ,        sin  C 

sin  A  =  sm  tb  -. — -:  =  sm  tp  -. =  k  sm  ^. 

^  sm  (p  sm  yu  ^ 

Whence 

cos  ^  =  Vi  —  sin'  ^  =  Vi  —  k^  sin'  ^. 
In  the  same  manner 

cos  ^  =  Vi  —  s\rfB  =  \/\  —  k"  sin'  0. 
Substituting  these  values,  we  get 

(2)  =^  +     ■  =  o. 

^  ^  Vi-k-"  sin'  0  '    V^i  -  y^'  sin'  ^ 

Integrating  this,  there  results 

r  defy  ,       /•*  ^^ 

(3)  /  =  +    /  — ^ =  =  const. 

7o    '^^l  -  K  sin'  0       y„    i^i  -  /^'  sin'  ^ 

When  0  =  o,  we  have  ^'  =  //,  and  therefore  the  constant 
must  be  of  the  form 


r  defy 

J,   Vi  -/^'sii70' 


i8 


ELLIPTIC  FUNCTIONS. 


whence 


(4) 


or 


/^'  sin'  0     y^   -/ 1  _  ^-^  sin'  ^ 


k"^  sin'  0' 


and  evidently  the  amplitudes  0,  ^',  and  jx  can  be  considered  as 
the  three  sides  of  a  spherical  triangle,  and  the  relations  between 
the  sides  of  this  spherical  triangle  will  be  the  same  as  those 
between  0,  ip,  and  yu. 

But  the  sides  of  this  triangle  have  imposed  upon  them  the 
condition 


sin  C 
sin  jJ. 


=  k; 


and  since  ^  <  r,  we  must  have  fx^  C,  which  requires  that  one 
of  the  angles  of  the  triangle  shall  be  obtuse  and  the  other  two 
acute. 

In  the  figure,  let  C  be  an  acute  angle  of  the  triangle  ABCy 
and  PQ  the  equatorial  great  circle  of  which  C  is  the  pole. 

The  arc  PQ  will  be  the  measure  of  the  an- 
gleC 

Let  AG  and  AH  be  the  arcs  of  two  great 
circles  perpendicular  respectively  to  CQ  and 
CP.  They  will  of  course  be  shorter  than  PQ. 
Hence  AB  =  jx  must  intersect  CQ  in  points 
between  CG  and  HQ,  since  jx>  [C  =  PQ).  In 
any  case  either  A  or  B  will  be  obtuse  according 
as  B  falls  between  QH  or  CG  respectively ;  and 
the  other  angle  will  be  acute. 

In  the  case  where  C  is  an  obtuse  angle,  it  will  be  easily  seen 
that  the  angle  at  A  must  be  acute,  since  the  great  circle  AD^ 
perpendicular  to  CP,  intersects  PQ  in  D,  PD  being  a  quadrant. 
The  same  remarks  apply  to  the  angle  B.     Hence,  in  eithet 


ELLIPTIC  FUNCTIONS. 


19 


case,  one  of  the  angles  of  the  triangle  is 
obtuse  and  the  other  two  are  acute,  as 
a  result  of  the  condition 

sin  C 

=k<\. 

sin  /i 

From  Trigonometry  we  have 

cos  fx  =  cos  0  cos  ^  -f-  sin  0  sin  ^  cos  C\ 

and  since  the  angle  C  is  obtuse, 


cos 


C  ^  —  ^\  —  sin'  C  =  —  Vi  —  /(''"'  sin^  yw, 


and 

(5)        cos  ^  =  cos  0  cos  ^'  —  sin  0  sin  ^'  Vi  —  k^  sin'  yw, 

the  relation  sought. 

The  spherical  triangle  likewise  gives  the  following  relations 
between  the  sides : 


<5)^ 


cos  0  =  COS  /<  cos  ^  -|-  sin  yu  sin  tp  Vi  —  ^''  sin"  0 ; 


(  cos  '/'  :=  cos  //  cos  0  -[-  sin  /^  sin  0  4^1  —  ^  sin"  ^. 
These  give,  by  eliminating  cos  M, 

cos"  4"  —  cos"  0 
"~  sin  0  cos  ip  J  ip  —  sin  ^  cos  (p  A  ^'' 

which,  after  multiplying  by  the  sum  of  the  terms  in  the  de- 
nominator and  substituting  cos"  =  i  —  sin",  can  be  written 

(sin*  0  —  sin"  f/?)(sin  0  cos  xp  A  ^  -\-  sin  ^'  cos  0/^0 
~~  sin"  0  cos"  rp  A^  rp  —  sin"  ^  cos"  (p  A"^  cp 

Since  the  denominator  can  be  written 

(sin"  0  —  sin"  ^)(i  —  k"  sin"  0  sin"  tp), 
^^  .  sin  0  cos  tp  A  tp  4-  sin  ip  cos  (p  A  d> 

<6)  Sm    W  =  ,a     .    a--. — ^j— . 

^  ^  ^  I  —  >^  sm   0  sm   ^ 

In  a  similar  manner  we  get 


r 


<6)* 


cos  /<  = 
A  tx  = 


cos  0  COS  ?/?  —  sin  0  sin  ip  A  ^  A  ip 
I  —  /^"  sin"  0  sin"  tp  ' 

A  (p  A  Ip  —  /6"  sin  0  sin  ^  cos  0  cos  ^ 
I  —  ^"  sin"  0  sin"  tp 


20  ELLIPTIC  FUNCTIONS. 

These  equations  can  also  be  written  as  follows 


sin  am  («  ±  k)  = 
{j\  -    cos  am  («  ±  J/)  = 
A  am  (m  ±  J')  = 


sin  am  «  cos  am  J'  /^  am  v  ±  sin  am  v  cos  am  u  A  am  u 


I  — 

-^■^ 

sin"''  am 

M  sin"''  am 

J' 

cos  am 

ti  cos  am  v 

T 

sin  am  u  sin  am  v 

A 

am  «  -i^ 

am  V 

I  — 

/&-^ 

sin'^  am 

u  sin^  am 

V 

^am  M 

A 

am  V  T 

K^  sin  am  w  sin  am  v  cos  am  «  cos  am  v 

or 


(8) 


I  —  k'  sin'-'  am  «  sin"''  am  v 

sx\  u  cx\  V  dnv  -^  sn  V  cn  u  dn  u 

sn  (m  ±  v)  = 

en  {ji  ±:  y)^= 

dn{u±v)=  ^  _  ^.  ^^,  ^^  ^^,  ^ 

Making  /^  =  r,  we  get  from  the  upper  sign 


I  - 

-  >^^  sn= 

u 

sn'^ 

V 

> 

en 

u 

en  V 

^  sn  z/ 

sn 

V  C 

n 

u 

dn  y 

I  - 

-  /^'  sn'^ 

z^ 

sn'^ 

y 

> 

dn 

u 

dn  r  q=  ^^  sn  ti  sn 

V 

en 

u 

en 

V 

(9) 


sn  2z^  = 


en  2u  = 


dn  2?^  = 


2  sn  //  en  z^  dn  « 
I  —  k"  sn*  z/     ' 

en"  ii  —  sn"  ti  dn'  7/         i  —  2  sn'  z^  H-.-^"  sn*  u 
I  —  ^'  sn*  z^  ~  I  —  k''  sn*  ?^  ' 

dn"  11  —  k^  sn"  u  en"  7/       i  —  2^"  sn"  z^  -[-  >^"  sn*  u 


From  these 


(lO) 


I  —  k^  sn*  u 


I  —  en  2u  = 


I  —  ^"  sn*  u 


2  en"  u  dn"  ?/ 
I  —  k'  sn*  /^ ' 


2  en   tc 

I  +  en  2u  =  :; r^ — «— ; 

'  I  —  k  sn  7/ ' 


I  —  dn  7^ 


2k^  sn"  7/  en'  u 
I  —  k''  sn*  u    * 


2  dn"  M 

I  +  dn  7^    =  >2 — J —  ; 

'  I  —  ^  sn  7^ ' 


ELLIP  TIC  F  UNC  TIONS. 


21 


and  therefore 


(II) 


en  2u 


sn  u  = 


en   u  = 


dn'  u 


I  -\-  dn  2u' 
dn  2u  -\-cr\2u 


I  -|-  dn  221     ' 

I  —  k''  -{-  dn  2u  -^  k^  en  2u 
I  -}-  dn  2« 


and  by  analogy 


u  /i  —  en  u 

2~  \J  \  -\-  d\\  ii' 


(12) 


u  /en  u  -f-  dn  « 

^"   2  ^  Y      I  +  dn  «    ' 

dn  -  =  A  /  ~ 

2         V 


k^  -\;-  dv\.  u  A^  k^  cn  u 
I  -|-  dn  « 

In  equations  (7)  making  u  =  v,  and  taking  the  lower  sign, 
we  have 

'  sn  0  =  0; 
(13)  -   en  0=  I  ; 

dn  o  =  I. 
Likewise,  we  get  by  making  u  =0, 

sn  (  —  «)  =  —  sn  « ; 


(14) 


en  (  —  7/)  =  -j-  en  «  ; 
dn  ( —  ti)  =  dn  u. 


CHAPTER   III. 
'  PERIODICITY  OF  THE   FUNCTIONS. 
When  the  elliptic  integral 

d<t> 


f. 


,     Vi  -  k'  sin'  0 

has  for  its  amplitude  -,  it  is  called,  following  the  notation  of 
Legendre,  the  complete  function,  and  is  indicated  by  K,  thus : 


„     Vi  —  /^'  sin=  0 

When  k  becomes  the  complementary  modulus,  k' ,  (see  eq.  4^ 
Chap.  IV,)  the  corresponding  complete  function  is  indicated  by 
K',  thus: 


l/i-/^'"sin'0 
From  these,  evidently, 


am  {K,  k)  =  1,  am  {K\  k')  =  \. 


sn  {K,  k)  —\\ 
(I)  \  en  {K,  k)=o\ 

dn  {K,  k)  =  k'. 


PERIODICITY  OF   THE  FUNCTIONS. 


23 


From  eqs.  (7),  (8),  and  (9),  Chap.  II,  we  have,  by  the  sub- 
stitution of  the  values  of  sn  {K)  =  i,  en  {K)  =  o,  dn  {K)  =  k' , 

'sn  2K  —  o  ; 
(2)  -  en  2K  =  —  I  ; 

dn  2^=  I. 

These  equations,  by  means  of  (i),  (2),  and  (3)  of  Chap.  II, 


give 
(3) 


fsn  («  -f"  2^)  =  —  sn  «  ; 
en  («  4~  2A')  =  —  en  «  ; 
dn  {u  -j-  2K^  =  dn  ?/ ; 

and  these,  by  changing  ii  into  21  -\-  2K,  give 

'  sn  {u-\- /^)  =^  snu\ 

(4)  -|  en  {u  -\-  /^K)  =  en  z/ ; 

dn  {jc  -\-  /^K)  =  dn  u. 

From  these  equations  it  is  seen  that  the  elliptic  functions 
sn,  en,  dn,  are  periodic  functions  having  for  their  period  4K. 
Unlike  the  period  of  trigonometric  functions,  this  period  is  not 
a  fixed  one,  but  depends  upon  the  value  of  k,  the  modulus. 

From  the  Integral  Calculus  we  have 


7.   ^^~J.  ^^^J^-  '^O'^J.  ^^^'"^J. 


-J: 


d(f) 

'A4> 


tiK; 


from  which  we  see  that 


7t 


n-  =  am  {nK') ; 
2 


24  ELLIPTIC  FUNCTIONS. 

or,  since  —  =  am  A, 

am  {nK)  =  n  .  am  K^ 
and 

nTT  =  am  {2nK), 

and  also  uTt  =  2n  am  TsT. 

In  the  case  of  an  Elliptic  Integral  with  the  arbitrary  angle  «', 
we  can  put 

Tt 

where  /?  is  an  angle  between  o  and  -,  the  upper  or  the  lower 

Tt 

sign  being  taken  according  as  -  is  contained  in  a  an  even  or 

an  uneven  number  of  times. 
/      In  the  first  case  we  have 


or,  putting  0,  =      —  nrt, 


rz'-^^rn: 


In  the  second  case 


*y  0  ^        c/»  ^        %J  nit- 


mt—^ 

or,  putting  (p^^=z  mt  —  0, 


defy 
J0' 


j--t=.„K-ri-L, 


PERIODICITY  OF   THE  FUNCTIONS.  25 

or  in  either  case, 


/ 


Thus  we  see  that  the  Integral  with  the  general  amplitude 
a  can  be  made  to  depend  upon  the  complete  integral  K  and 

an  Integral  whose  amplitude  lies  between  o  and  — . 


Put  now 


This  gives 


—r^  =  u,  0  =  am  u. 


d<P 


or  am  {2nK  ±_u)  ■=  nn  ±_  fi 

(5)  =  rnt  ±i  am  u 

(6)  =  2n  .  am  /T  ±  am  « ; 
or,  since                  am  (—  z)  ■=  —  am  2, 

am  {u  ±  2nK)  =  am  u  ±  nyr 

=  am  u  ±.  2n  .  am  TT. 

Taking  the  sine  and  cosine  of  both  sides,  we  have 

sn  {u  -\-  2TiK)  =r  ±  sn  ?/  ; 
en  {u  -\-  2nK)  =  ±  en  ?/ ; 

the  upper  or  the  lower  sign  being  taken  according  as  n  is  even 
or  odd.     By  giving  the  proper  values  to  n  we  can  get  the  same 
results  as  in  equations  (3)  and  (4). 
Putting  n—  I  in  eq.  (5),  we  have 

sn  {2K  —  ti)  =  sm  7t  en  u  —  cos  ;r  sn  « 

(7)  =  sn  u. 


26 


ELLIPTIC  FUNCTIONS. 


Elliptic  functions  also  have  an  imaginary  period.     In  order 
to  show  this  we  will,  in  the  integral 

J.  ^'t'' 

assume  the  amplitude  as  imaginary.     Put 

sin  (p  ^=  i  tan  ^|}. 
From  this  we  get 

cos  0  = 


(8) 


cos  Ip  ' 
-     Acp  =  _^i  -^"sin^^  _  ^ii,,  k') . 


cos  ip 


cos  Ip 


d(p  =  t 


.    dtp 


cos  rp  ' 
From  these,  since  0  and  rp  vanish  simultaneously,  we  easily  get 


d^  _  .    r^      dip 
//0-V      A{tp,kr 


Put 


whence 


^  -  =  u     and     ^  =  am  {u,  k'\ 


^  iu     and     0  =  am  iiii)  ; 


and  these  substituted  in  Eq.  (8)  give 

sn  iu  =  i  tn  {u,  k')  ; 


(9) 


cr\  tu  =  ~ 


dn  iu  = 


en  {ji,  k')  ' 

dn  {u,  k') 
en  («,  /^')  * 


PERIODICITY  OF    THE  FUNCTIONS. 


27 


By  assuming 


'•^       dij: 


J{iP,  k') 


jT  =  m      and       /     -r- 


J.  2-^=-"' 


we  gfet 


sn  (—  ti)  =  i  tn  {iu,  k'), 
I 


en  (—  u)  = 


en  {iu,  k') ' 


dn  (tu,  k') 
dn  (—  u)  =  — J-. — y/  ; 
^        ^       en  {lu,  k)  ' 


or,  from  eq.  (14),  Chap.  II, 

sn  u  =  —  /  tn  {m,  k') ; 

I 


(10) 


en  u  = 


dn  u  = 


en  {iu,  k')  ' 
dn  {iu,  k') 


en  {iu,  k') ' 

From    eqs.   (7),   Chap.    II,  making  r  =  K,  we  get,  sinee 
sn  K  =  I,  en  K  =  o,  dn  K  =  k', 


(II) 


sn  {u  ±  K) 


en  {u  ±K)  — 


dn{u±K)=^-^ 


en  u  dn  u 
\  —  k^  sn*  u 

T  sn  «  dn  uk' 
dn"  u 

k' 


=  =F 


en  ti 
dn  «  ' 

k'  sn  « 
dn  u   ' 


In  these  equations,  ehanging  u  into  /«,  we  get,  by  means  of 
eqs.  (9), 

sn  (iu  ±  K^  =  ± 


(12) 


en  {iu  ±  K)=T 
dn  {iu  ±K)  —  ^ 


dn  {u,  k') ' 

ik'  sn  {u,  k') 
dn  (?^  /^')    ' 

/^'  en  («,  ^') 


dn  {u,  k') 


28 


ELLIPTIC  FUNCTIONS. 


Putting  now  in  eqs.  (9)  u  ±  K'  instead  of  u,  and  making  use 
of  eqs.  (10),  and  interchanging  k  and  k' ,  we  have 


(13) 


sn  {ill  ±  iK'^  =  — 


z  en  {ti,  ^') 
k  sn  {ii,  k') ' 


^  ^  iC  sn  {z(,  k) 

dn  {iu  ±  iK)  =  qp 


sn  {H,  k') 


Substituting    in    these  —  iu    in    place    of   u,   we    get,    by 
means  of  eqs.  (9)  and  eqs.  (14)  of  Chap.  II, 


(14) 


sn  {ii  ±  iK')  =  T" 


sn  u 
i  dn  u 


cn{u±  iK)  =  =F  ,  ^„  „ , 
dn  {u  ±  iK)  =  '^f  i  cot  am  u. 


In  these  equations,  putting  u  -\-  K  in  place  of  u,  we  get 
sn  {u-}-K±  iK)  =  +  v^'^  ; 

^      '  ^  k  cn  u 

dn{2{-^K±  iK)  =  ±  ik'  tn  u. 

Whence  for  «  =  o  we  get 


(16) 


sn  {K  ±  iK)  =  p 

cn{K±iK)  =  ^'^', 
dn  {K  ±  iK)  =  o. 


PERIODICITY  OF   THE  FUNCTIONS.  2^ 

If  in  eqs.  (14)  we  put  z/  =  o,  we  see  that  as  u  approaches 
zero,  the  expressions 

sn  (±  iK'\     en  (±  iK'\     dn  (±  iK) 

approach  infinity. 

We  see  from  what  has  preceded  that  Elh'ptic  Functions 
have  two  periods,  one  a  real  period,  and  one  an  imaginary 
period. 

In  the  former  characteristic  they  resemble  Trigonometric 
Functions,  and  in  the  latter  Logarithmic  Functions. 

On  account  of  these  two  periods  they  are  often  called 
Doubly  Periodic  Functions.  Some  authors  make  this  double 
periodicity  the  starting-point  of  their  investigations.  This 
method  of  investigation  gives  some  very  beautiful  results  and 
processes,  but  not  of  a  kind  adapted  for  an  elementary  work. 

It  will  be  noticed  that  the  Elliptic  Functions  sn  w,  en  u,  and 
dn  u  have  a  very  close  analogy  to  trigonometric  functions,  in 
which,  however,  the  independent  variable  ti  is  not  an  angle,  as 
it  is  in  the  case  of  trigonometric  functions. 

Like  Trigonometric  Functions,  these  Elliptic  Functions  can 
be  arranged  in  tables.  These  tables,  however,  require  a  double 
argument,  viz.,  ii  and  k.  In  Chap.  IX  these  functions  are  de- 
veloped into  series,  from  which  their  values  may  be  computed 
and  tables  formed. 

No  complete  tables  have  yet  been  published,  though  they 
are  in  process  of  computation. 


CHAPTER   IV. 
LANDEN'S  TRANSFORMATION. 


Let  AB  be  the  diameter  of  a  circle, 
with  the  centre  at  O,  the  radius  AO  ^  r, 
and  C  a  fixed  point  situated  upon  OB, 
and  OC  =  kj.  Denote  the  angle  PBC 
3  by  0,  and  the  angle  PCO  by  0j.  Let 
P'  be   a  point  indefinitely   near   to   P. 


PP'       sin  PCP       sin  PCP 


PC  ~  sin  PP'C  ~~  cos  OPC 

But     PF  =  2rd(p,     and     s\n  PCP' =  PCP' -  dcj)^-, 

therefore 

2r(^0  dcf)^ 

TC'  ^  ^^ToFC 
But 

PC^  =  r"  +  r'4=  +  2r'4  cos  20 

=  (r  +  r^„)*  cos'^  0  +  (^  -  ^^o)'  sin'  0; 
also  r"  cos"  CP'C"  =  r^  -  r^  sin=  OF'C 

^r"  -  r'k^  sin'  0,. 
Therefore 

2(3^0  _  </0, 


V(r  4-  rk^'  cos'  0  +  (r  —  r/^„)  sin'  0       -kV'  —  r'>^;  sin'  0/ 
which  can  be  written 


^0 


^  +  ^^»|/j__4^ 


7 — i — TTzSm   0 


30 


.-«rt)C"-.;'Ji-'-i*A-; 


LAN  DEN'S    TRANSFORMATION.  3 1 

Putting 

we  have 

^'-^        ^0      ^^^'^''  sin*  0  2     ^^      |/i  -  /^;  sin'  0. ' 

no  constant  being   added    because  0  and   0,  vanish  simulta- 
neously ;  0  and  0,  being  connected  by  the  equation 

s\r\  OPC      sin  (20  — 0,)       rk^ 
^3)  sin  (96'/'  ^       shr0;^       =  "7  ^    •• 

From  the  value  of  k'  we  have 

(4)  l-k  =k    =(7Tp^)'» 

and  therefore 

I  —  k' 

(5)  ^•  =  r+T- 

k'  is  called  the  complementary  modulus,  and  is  evidently   the 
minimum  value  of  J0,  the  value  of  A(}>  when  0  =  90°  : 


From  eq.  (i)  we  evidently  have  k  >  k^,  for,  putting  eq.  (i) 
in  the  form 

^  _  4 

we  see  that  if  ^„  =  i,  then  >^  =  ^„,  but  as  ^^  <  i,  always,  as  is 
evident  from  the  figure,  k  must  be  greater  than  k^. 

It  is  also  evident,  from  the  figure,  that  0,  >  0.  Or  it  may 
be  deduced  directly  from  eq.  (3). 

Since  k  <  i,  we  can  write 

k  =  sin  e,  k'  =  Vi  —  k^  =  cos  e. 


32  ELLIPTIC  FUNCTIONS. 

Substituting  in  eq.  (5),  we  have 

and  we  can  write 


k,  =  sin  e„         k/  =  \/i-k:  =  cos  e, . 

From  eq.  (5)  we  have 

1+^0=       ^ 


I  +/&'• 
Substituting  the  value  of  k^  in  that  for  k/,  we  get 

"'  -  i-{-k" 
We  also  have 

20  —  0,  =  0  —  (0.  —  0) 

and,  eq.  (3),  sn  (20  —  0,)  =  k,  sin  0^ , 

becomes 

sin  0  cos  (01  —  0)  —  cos  0  sin  (0,  —  (p)  = 

k,  sin  0  cos  (0,  —  0)  +  '^o  cos  0  sin  (0,  —  <*), 
or 

tan  0  —  tan  (0,  —  (p)  =  k,  tan  0  +  /^o  tan  (0,  —  0), 

or 

1  —  k 
tan  (0,  —  0)  =  j-qr;^  tan  0 

=  k'  tan  0. 


Collecting  these  results,  we  have 

2  VI. 


2vk,       .   ^ 

(6)  ^=^-p-^-  =  sm^; 

I  —  k! 

(7)  K  =  TXT'  =  ^'"  ^»  =  ^^^^  ^^ ' 


LANDEN'S    TRANSFOJiMATION.  33 


(8) 

(9) 

(ro) 


k: 

2VW 
~  \-\-k' 

=  COS  d^ ; 

k' 

=  cos  0 ; 

^-\-K 

2 

~     k     ~ 

k: 

~   \^k'' 

A/k' 

4/p  "~  cos"  ^d  * 

(11)  sin  (20  —  0,)  =: /^„  sin  0,; 

(12)  tan  (01  —  (p)  =  k'  tan  0 ; 

^0. 


^(^0,  0.)  ' 


(14)  >^  =  |/i  -  /^'»,    k'  =  Vi-  k\ 

Upon  examination  it  will  easily  appear  that  k  and  k^ ,  and  0 
and  d^ ,  are  the  first  two  terms  of  a  decreasing  series  of  moduli 
and  angles;  k'  and  k^ ,  and  0  and  0,,  of  an  increasing  series; 
the  law  connecting  the  different  terms  of  the  series  being  de- 
duced from  eqs.  (6)  to  (12). 

By  repeated  applications  of  these  equations  we  would  get 
the  following  series  of  moduli  and  amplitudes : 


K  ^:  0. 

k  k'  xl> 

The  upper  limit  of  the  one  series  of  moduli  is  i,  and  the 
lower  limit  of  the  other  series  is  o,  as  is  indicated,     k  and  k\ 


34 


ELLIPTIC  FUNCTIONS. 


which  are  bound  by  the  relation  k^  -\-  k'^  =z  i,  are  called  the 
primitives  of  the  series. 

Note. — It  will  be  noticed  that  the  successive  terms  of  a  decreasing  series 
are  indicated  by  the  sub-accents  o,  oo,  03,  04,  .  .  .  on;  and  the  successive  terms 
of  an  increasing  series  by  the  sub-accents  i,  2,  3,  .  .  .  n. 

Again,  by  application  of  these  equations,  we  can  form  a  new 
series  running  up  from  k,  viz.,  k^,  k^,  k^,  .  .  .  k„=^  i(«=co)  ;  and 
also  a  new  series  running  down  from  k',  viz.,  ^/,  k\^,  .  ,  .  k^„=. 
O(„=oo)-     So  also  with  0. 

Collecting  these  series,  we  have 

Kn  =  0  kj  =   1  0„ 


k 

k 
h 


0, 

000 


kn=l  k\„  =  0  0o«  =  O 

Note. — In  practice  it  will  be  found  that  generally  «  will  not  need  to  be  very 
large  in  order  to  reach  the  limiting  values  of  the  terms,  often  only  two  or  three 
terms  being  needed. 

Applying  eqs.  (7),  (12),  (13),  and  (14)  repeatedly,  we  get 
=  sin  d,  k'  =  cos  6; 

i-k' 


(14.)   \ 


k.= 


-,,  =  tan'  ^6  =  sin  6^, 


I -i-k' 

k^^  =  tan'i^„  =:sin  6^^, 
k,,  =  tan'  ^d^,  =  sin  6^, , 


^    k,„  =  tan'  ^e,^„_,)  =  sin 


k/  =  cos  0^ ; 

k,'  =  cos  0,^ ; 
k'  =  cos  9.^ ; 


kj  =  cos  0^. 


LA  NDEN  •  S    TRA  NSFORMA  TION. 


35 


'  tan  (si,  —  0)  =  k'  tan  ^ ; 
tan  (0,  —  0J  =  kl  tan  0, ; 
(14,)  \   tan  (0,  -  0,)  =  k:  tan  0. ; 

^  tan  (0„  —  0„..,)  =  >^'(„_,)  tan  0„_,. 

/^/^,  0)=i±^»/^(>^„,0O; 


F{Kn-^)^  0«-.)  =  1-—  ^'^o«  ,   0»)- 


Multiplying  these  latter  equations  together,  member   by 
member,  we  have 

(15)     F{k,  0)  =  (I  +  ^„)(i  +  /&.„) . . .  (I  +  K)  ^%^"^ ; 

A»  '^00  >  etc.,  and  0,,  0,,  etc.,  being  determined  from   the  pre- 
ceding equations. 

From  eqs.  (9)  and  (10)  we  get 

^  +  '^''  -  c^Tl^'       I  +  '^oo  -  ^,^5,  ^Q^ ,     etc. 


Substituting  these  in  eq.  (15),  we  get 

I  F(k,„,  <p„) 


(16)         Fik,<p)  = ^ 


e. 


m  Q       0  3      on 

cos'  —  cos  —  ...  cos  — 

2  2  2 


36  ELLIPTIC  FUNCTIONS. 

From  eqs.  (15)  and  (10)  we  get 


And  this  with  equations  (8)  and  (9)  gives 


.     .             rrru    ^\            /COS  8,  COS  0„„  .  .  .  cos'  6/„„     F{k„„ ,  0„) 
(17)  F(/^,  0)  =  y ^^^-^ ^;i . 

Applying  equation  (13)  to  {k^ ,  0„),  {k,,  cpj,  etc.,  we  get 


but  since,  eq.  (lo), 

i+k  I 


-/ ,  etc., 


2  I  +  /^„ 

these  become 


F{k„_, ,  0„(„.,))  =  (I  +  ^\nW{k,„  0„„) ; 

whence 

(18)    F[k,  0)  =  (I  +^oO(i  +  ^'00)  .••(!  +  ^\n)Fik„,  cp,„), 

in  which  >^/,  /&'„„,  etc.,  /&,  ,/^,,  etc.,    0„,  0„„ ,  etc.,    are  deter- 
mined as  follows  : 

Let  ^  =  sin  ^, 

/^j  =  sin  ^j . 

From  eq.  (10), 

2Vk  .     /,         2  i^sin  ^ 

k,  =  — r~L    or     sm  0^  =  -- ^ — ; — a- 
'       I  -|-  >^  I  -|-  sm  t' 


(i8J 


LA  NDEN'  S    TRA  NSFORMA  TION. 
Solving  this  equation  for  sin  0,  we  get 

sin  e  =  tan'  ^0^ . 
Hence  we  can  write 

f  /^  =  sin  ^  =  tan'  i^. ; 
k^  =  sin  ^j  =  tan"  ^6'^ ; 


37 


(1 8,) 


^  k„  =  sin  B„ . 
From  equation  (12)  we  get 

sin  (20,   —  0)  =  >^  sin  0  ;  * 
sin  (20„„  —  0„)  =  k,  sin  0„ ; 


sin  (20„„  —  0„(„_x))  =  /^„-i  sin  0„(„_„. 


*  When  sin  0  =  i  nearly,  0  is  best  determined  as  follows:  From  eq.  (12) 
we  have 


whence 


tan  (0  —  0o)  =  ^0'  tan  0o 

=  k^  tan  0  nearly; 

0  —  00  —  i^-^o'  tan  0  nearly, 


R  being  the  radian  in  seconds,  viz.  206264". 806,  and  log  ^=  5.3I4425I. 

Substituting  the  approximate  value  of  0o ,  we  can  get  a  new  approximation. 


Example. 


k\a  =  log-i  5.8757219 


2.0707179        117". 684  =  i'.96i4 

<po  —  000  ^=  l'.96l4 

000  =  82°  28'.o386  1st  approximation. 


00  = 

82° 

30' 

tan 

82°  30' 

10.8805709 

i  00 

5-8757219 

iP 

5-3Mt25i 

38  ELLIPTIC  FUNCTIONS. 

To  determine  k^ ,  k\^ ,  etc.,  we  have 


r    /./ 


=  sin  7/,  k  =  cos  7 ; 

I  -^ 

I   ^'oo  =  tan'  ^7„  =  sin  77„„ ,  /&^  =  cos  t;^,  ; 

t  etc.  etc.  etc. 

Or,  since, +4' =3^^,     ,  +  ^'..  =  _1^,  etc., 

we  can  put  eq.  (i8)  in  the  following  form  : 

( .9)       F{i.  <p)  =  ,os-  i„  cos-  i„,...  cos-  inJ^^' '  «• 
From  equation  (13)  we  have 

(19)*  F{k„<p,)='~^F{k,<p\ 

whence 

By  repeated  applications  this  gives,  after  combining, 

^(*. -!*)  =  r:^  ■  i^;  •  • -rif^;  >^(^.,  0„) 


This  value  gives 

00  —  000  =  117". 1675  =  l'. 95279 

.  ■ .     000  =  82°  28'. 04721  2d  approximation. 

This  value  gives 

00  —  000  =  117". 1698  =  i'.95283 

000  =  82°  28'. 04717  3d  approximation. 


LANDEN'S    TRANSFORMATION.  39 


(20)  m  0)  =  J^'  ^'  •;  •  ''•■'    .  F(i.  ,  0..)  ; 

k^,  k^,  etc.,  being  determined  by  repeated  applications  of 

M    2\rk 

or  by  equations  (18,). 

In  equation  (19)*  let  us  change  k^  and  0^  into  k'  and  0  re- 
spectively, so  that  the  first  member  may  have  for  its  complete 
function 

K'  =  F{k',  0). 

Upon  examination  of  eq.  (19)*  we  see  that  the  modulus  in 
the  second  member  must  be  the  one  next  less  than  the  one  in 
the  first  member,  that  is,  kj;  and  likewise  that  the  amplitude 
must  be  the  one  next  greater  than  the  amplitude  in  the  first 
member,  viz.,  0,  ;  hence  we  get 

F{k',  cp)  =  '-±^'  F{k:,  0,). 

Indicating  the  complete  functions  by  K'  and  JC/,  we  have, 
since  0  =  -  when  (p^  =  tt  (see  Chap.  V), 

and  in  the  same  manner, 

^/  =  (l+^'oo)^'oo, 


^o\n-i)  —  (I   +  k\^K\^\ 


40  ELLIPTIC  FUNCTIONS. 

whence 

iT'  =  (I  +  k:\Y  +  i;j . . .  (I  +  k\;:^K\^, 

Since 


^'o«  —    }     ^^  —  ~2>  (^  =  limit,) 


we  have 

(20)*  (I  +  ^;)(i  +  k\:^ . . .  (I  +  .^'„„)  =  H^  . 

From  eq.  (19)*  we  have,  since  [eq.  (10),  Chap  IV] 
I  +/&  I 


(1  +  ^, 


2        ~I+>&o" 


j(0„,/^,)  ^^  40,,/^)  5 


whence  also,  since  for  0^  =  -,  0  =  ;r, 


(I  +  ^'„„)  K^  =  2K, , 


and 


or 


(I  +  /&/)(!  +  kj) . . .  (I  +  ^;„)a;  =  2-K', 

K^ ^ _ 


(21)  =^K, 


LANDEN'S    TRANSFORMATION.  4 1 

Let  US  find  the  limiting  value  of  F{k^^ ,  0„)  in  eq.  (15).     In 
the  equation  tan  (0„  —  0„-,)  =  >^„-,  tan  0„_i ,  we  see  that  when 
^„_i  reaches  the  limit  i,  then  0„  —  0„.i  =  0„_i  or  0„  =  20„_,. 
Therefore 

2"  2"  2""'  ' 

0>,+.  _  ^0»  —  ^?  —  ^!!Zi . 

-^^  =  -^  =  constant,  whatever  m  may  be. 
Therefore  eq.  (15)  becomes 

(21)*  F{k,   0)  =  (I   +  /^o)(l  +>^o.)  .  .   .  (I  +  >&.«)  1^, 

0, 
;«  being  whatever  number  will  carry  k^  and  — ^  to  their  limiting 

values. 

In  the  same  way,  eqs.  (16)  and  (17)  become 


(22) 


I 

0- 

,^ 

.^0 

,    ^on 

2" 

cos 

2 

cos 

2 

COS 

2 

./ 

cos 

^0 

COS 

^00 

.  .  . 

cos" 

^0„ 
0« 

0« 

V 

COS 

e 

2»' 

(23) 


72  —  I  being  the  number  which  makes  k'„_^  =  i. 

In  these  last  three  equation  k^ ,  ^„„  are  determined  by  eqs. 
(14,) ;  0, ,  0, ,  etc.,  by  eqs.  (14,)  *  ;  6,0^,  etc.,  by  eqs.  (14,) ;  and 
^\  k/,  k^,  etc.,  for  use  in  eq.  (14,)  by  eqs.  (14,). 

*  Taking  for  0i  —  (p,  etc.,  not  always  the  least  angle  given  by  the  tables, 
but  that  which  is  nearest  to  (p. 


42  ELLIPTIC  FUNCTIONS. 


BISECTED 

AMPLITUDES. 

We  have  identically 

u 

'  2  ~ 

■J 

r  " 

u 
am  — 

2 

/,- 

-  k'  sn= 

2 

u 
2  ~~ 

u 
2  .-  = 

4 
etc. 

2F[k, 

u\ 
,am-   ; 
4/ 

Therefore 

=  F{k,  am  u)  =  2"F[k,  am  — ) 


u 


=  2"  .  am-  ,  {n  =  limit,) 


ti 


am  —  being  determined  by  repeated  applications  of  eq.  (12)  of 


n 


Chap.  II,  as  follows : 


.  u      I  —  en  71      2  sin"  -i-  am  u 

sn  —  = ^ * 

2       I  -\-  dmi  I  -\-  dn  u     ' 

.    am  u 


.     -,  sm 

u  sm  f  am  u  2 

(24)  sn  -  _  -_^==  =  -^^^  ; 

2 
/?  being  an  angle  determined  by  the  equation 


(25)  cos  ^  =z  dr\  u  =^  Vi  —  k^  sn"  u, 

and  n  being  the  number  which  makes 

u 
2"  am  —  =  constant. 
n 

u 
am  -  is  found  by  repeated  applications  of  eq.  (24). 


LANDEN'S    TRANSFORMATION.  43 

Indicating  the  amplitudes  as  follows : 
am  u  =  0, 

am  --  =  (p,, , 


am-  =  0„,, 
4 


am7r  =  0o8» 


u 
am-„  =  0o,«,- 


(26)  /(/fe,  0)  =  2"0.,«; 

«  being  the  limiting  value. 

In  eq.  (i8),  when  k^  reaches  its  limit  i,  we  have 

F{K ,  0«„)  =    r  -~  =  log.  tan  (45°  +  i0„„) , 
^„     cos  <p„„ 

and  eqs.  (18)  and  (19)  become 

(27)  i^(/&,  0)  =  (I  +  ^:){i  +  >^'„„) . . .  (I  +  ^'J log.  tan(4S''+i0.») 

=  cos-  i^  cos'i.l  .  .  .  cosHVo.^"^'  tan(45°  +  M.) 
^^^^       *  =  cos"ivcos'K...cos-iv„„ '  |^^°g^^"(45°+i0o«); 


«  being  the  number  which  renders  k^  =  i, 
Eq.  (20)  becomes 


44  ELLIPTIC  FUNCTIONS. 

(29)  F{k,  0)  =      /^A^LlK  .  log.  tan  (45°  +  i0„„) 

^      /^A^_^. ^  log  tan  (45°  +  i0o«) 

^        /cos  ^0  cos  ;;„„...  COS>„„  _     I     i^g  ^^^  ^^50  _^  ^^^^)^ 

y  cos  T}  M 

In  these  equations  >^/,  >^'o„,  etc.,  are  determined  by  eqs.  (183); 
rf,  77„,  etc.,  by  eqs.  (183);  0„,  0„„,  etc.,  by  eqs.  (18,) ;  k,,  k^,  etc., 
by  eqs.  (18.). 

Substituting  in  eq.  (27)  from  eq.  (20)*,  we  have 

2K' 

F{k,  0)  =  — -  log.  tan  (45°  +  10.„) 

7t 
2K' 

(30)  =  ^^  log  tan  (45°  +  i0,„). 


CHAPTER  V. 
COMPLETE   FUNCTIONS. 
Indicate  by  K  the  complete  integral 

r^  d(t) 

and  by  K^  the  complete  integral 


(2) 


^•-7^     Vi  -  y^;  sin' 0, 


and  in  a  similar  manner  A'oo,  iir„3,  etc. 
From  eq.  (12),  Chap.  IV,  we  have 

tan  (0,  —  (p)  =  k'  tan  <p 

tan  0,  —  tan  0 


I  +  tan  0,  tan  0  * 


whence 


(i  +  k')  tan  0 
tan  0.  =  T^-^^l^^ 


k'  tan  0 


tan  0 

45 


46  ELLIPTIC  FUNCTIONS. 

From  this  equation  we  see  that  when  0  =  - ,  0^  =  ;r.    This 

same  result  might  also  have  been  deduced  from  Fig.  i,  Chap. 
IV,  or  from  the  equation 

(3)  0,  =  20  —  k,  sin  20  +  \k^  sin  40  —  etc., 

this  last  being  the  well-known  trigonometrical  formula 

tan  ;r  =  ;z  tan  y. 


\  —n 


,     l/l  —7lV  .  I/I  —  «v        , 


.r  =  r ; —  sm  _,    ,      ■      ,      ,  ^...  ^,         , 


Since  /  '  ^^  ^^  l^  =  i^T, ,  we  must  have 


/ 


^{kM 


These  values  substituted  in  eq.  (13),  Chap.  IV,  give  succes- 
sively 

<4)  ir=(i+^„)^., 

^o=(l+^oo)^„o, 


whence 

<5)  ^=  (I  +  ^„)(i  +  k:)  . . .  (i  +  /^„„)a;„. 

Since  the  limit  of  i„„  is  o,  K^^  becomes 


K^,.  — 


f :'".=--' 


COMPLETE  FUNCTIONS.  47 


and  we  have 


(6)  ^=^(i+/^.)(i+^J... 


(7) 


i^ 


cos'^  \B  cos'  *^„  .  .  .  cos"'  i6^„ 


>&,,  >^„,  etc.,  and  ^,,  ^o.  etc.,  being  found  by  eqs.  (14,)  of  Chap. 
IV. 

From  the  formulae  in  these  two  chapters  we  can  compute 
the  values  of  u  for  all  values  of  0  and  k  and  arrange  them  in 
tables.     These  are  Legendre's  Tables  of  Elliptic  Integrals. 


CHAPTER    VI. 

EVALUATION    FOR   <p. 

TO   FIND   (p,   ?/   AND   k   BEING   GIVEN. 

From  eqs.  (21)  and  (23),  Chap.  IV,  we  have  {n  having  the 
value  which  makes  cos  0^„  =  i) 


2"U  2"U  I^COS  6 

<^)    ^«  =  (I  +  k,){i  +  k,,)  .  .  .  (I  +  K)  -   4.'cos  e^...  cos^  6  J 

from  which  0„  can  be  calculated,  k^,  k^^,  etc.,  being  found  by 
means  of  equations  (14,),  Chap.  IV. 

Then,  having  0„ ,  k„ ,  k^^ ,  etc.,  we  can  find  0  by  means  of 
the  following  equations: 

sin  {2(p„_,—  0„)  =  k,„  sin  <p„, 
sin  (20„_2  —  0„_,)  =  k„^„.,)  sin  (p„_^ , 


sin  (20  —  0,)  =  k^  sin  0, ; 

whence  we  can  get  the  angle  0. 

When  k>  V^  the  following  formulae  will  generally  be  found 
to  work  more  rapidly  : 

From  eq.  (29),  Chap.  IV,  we  have 

(2)  log  tan  (45°  +  i0„„)  =  -7-,  , 

^  k 

48 


EVALUATION  FOR   (p.  49 

from  which  we  can  get  0„„ ;  k^,  k^,  etc.,  being  calculated  from 
eqs.  (i8,),  Chap.  IV,  and  0  being  calculated  from  the  follow- 
ing equations : 

tan  (0,(„-,)  —  0o»)  =  K  tan  0„„, 

tan  (0„  —  0  J  =  >^,  tan  0„, 
tan  (0  —  ({)^)  =  k  tan  0, ; 

whence  we  get  0. 

This  gives  a  method  of  solving  the  equation 

Ftp  =  nF(t>, 

where  n  and  0  and  the  moduli  are  known,  and  ^  is  the 
required  quantity.  7i  and  0  give  i^//',  and  then  ^  can  be  deter- 
mined by  the  foregoing  methods. 

When  ^  =  I  nearly,  equation  (2)  takes  a  special    form, — 

1°.  When  tan  0  is  very  much  less  than  p-.     In  this  case 


d4> 


^^'  *^)      J    |/cos^  0  +  k'^  sin"  0     J    V{i-\-  k"- 


tan'  0)  cos*0 


=y,-^  =  logtan(45°  +  i0); 


cos  0 
whence  we  can  find  0. 


I 


2°.  When  tan  0  and  -77  approach  somewhat  the  same  value, 

and  k'  tan  0  cannot  be  neglected,  F{k,  0)  must  be  transposed 
into  another  where  k'  shall  be  much  smaller,  so  that  k'  tan  0 
can  be  neglected. 


50  ELLIPTIC  FUNCTIONS. 

Tt 

These  methods  for  finding  0  apply  only  when  0<  -,  that 
is,  u  <  K.     In  the  opposite  case  {u>  K)  put 

u  =  2nK  ±  V, 

the  upper  or  the  lower  sign  being  taken  according  as  K  is  con- 
tinued in  u  an  even  or  an  odd  number  of  times.     In  either  case 
y  <  K,  and  we  can  find  v  by  the  preceding  methods. 
Having  found  v,  we  have  from  eq.  (5),  Chap.  Ill, 

am  u  —  am  {inK  ±  v) 
=■  HTC  ±.  am  V. 


CHAPTER   VII. 
DEVELOPMENT   OF   ELLIPTIC  FUNCTIONS  INTO  FACTORS. 
From  eq.  (12),  Chap.  IV,  we  readily  get 
sin  (200  —  0)  —  >^  sin  0 ; 

sin  200 


sin  0  = 


Vl  4-  >^'  +  2k  COS  20„ 

sin  200 
l/(i+/^»)-4/^sin^  0„ 

I  _[-  k^  si"  20„ 

2~    '    i^i  -k^  sin'  0; 


\^since  — — r  —k^  and  i-|-/^=     ,  ,  ,,  eqs.  (6)  and  (10),  Chap.  IV j ; 
and  thence 

{l)  sin  0 


and  thence 

(i  4->^/)sin  0„cos0, 


^(00,/^.) 

From  eq.  (13),  Chap.  IV,  we  have 

^0„  I  +  /^    r*     ^0 


^(0„,/&,)  2       /     ^(0,  >^)' 

and  from  eq.  (4),  Chap.  V,   passing  up  the  scale   of  moduli 
one  step, 

I  -f-  ^  -  ^, 

51 


52  ELLIPTIC  FUNCTIONS. 

whence 

Put 

^(00  >  A)  =  ^^     and     F{(}),  k)  =  u, 
whence 

^. 

^'  =  ^«- 

Furthermore, 

0  ^  am  (z^,  >^) ; 

0,  =  am  (z^, ,  k;)  =  am  [7^^^  >^J- 

Substituting  these  values  in  eq.  (i),  we  have 

sn  [^n,^)  en  (-^^^-^aj 
sn  («,  k)  =  {i-\-  k,')  — 


dn  (^^^  k) 


But  from  eq.  (ii),  Chap.  Ill,  we  have 

en  {v,  k^  (     \    ir     h\ 


or 


en  (--^^^  /^a)  /A-z^  \ 


sn  ,      „ 
2K 


dn  (2":^^''  ^J 


DEVELOPMENT  INTO  FACTORS. 


53 


whence 

(2)      sn  (./,  /^)  =  (I  +  K)  sn  §  sn  [^  (^  +  2K)\  "^ 

(Mod.  =  /&,.) 
From  this  equation,  evidently,  we  have  generally 

(2)*  sn  (v,  ^„)  =  (I  +  /^;,„^.,)  sn  ^  V  sn  [^  (v  +  2A'„)]- 

(Mod.  =  >^«  +  ,.) 

Applying  this  general  formula  to  the  two  factors  of  eq.  (2), 
we  have 

(Mod.  i,) 


sn 


(^,>^.)  =  (l  +  ^'o„) 


sn  —75^  •  — 7^  .  sn 
2ir,       2K 


=  (i  +  >^'.o)  sn  ^  sn  ^  («  +  ^K) ;     (Mod.  /&, ;) 
(3)       sn  [^-  («  +  2K),  /^.]  =  (I  +  ^'00)  sn-^  {u  +  2^^) 

^[^  («  +  2/^)  +  27r.].         (Mod.  K 0 


.  sn 


2K. 


The  last  argument  in  this  equation  is  equal  to 

and  since,  eq.  (7),  Chap.  Ill, 

sn  {u,  k^  =  sn  {2K^  —  u,  k^), 

f  The  analogous  formula  in  Trigonometry  is 

sin  0  =  ^  sin  ^0  sin  ^(0  -|-  *). 


54  ELLIPTIC  FUNCTIONS. 

we  can  put  in  place  of  this, 


2'K 

whence  eq.  (3)  becomes 


sn 


K, 


K. 


(4) 


•  sn  p^  {2K  -  u).  (Mod.  /&, .) 

Substituting  these  values  in  eq.  (2),  we  have 

s"(«'^)  =  (i+/^oO(i+^'o„rsng 
K  K 


in  which  the  double  sign  indicates  two  separate  factors  which 
are  to  be  multiplied  together. 

By  the  application  of  the  general  equation  (2)*  we  find  that 
the  arguments  in  the  second  member  of  eq.  (4)  will  each  give 
rise  to  two  new  arguments,  as  follows : 


and 


Yk    ^^^^^  iW 

(Ku  \        K 


K,(Km 

2K. 


K  K 

■^{2K±u)    gives    -^{2K±u\ 


DEVELOPMENT  INTO  FACTORS.  55 

and 

^  [^  {2K  ±  «)  +  2K^  =  ^  (loATi  «),  .     .  {a) 

K  K 

-^  {aK  +  u)    gives    ^  (4Ar  +  «), 


and 


5§-  [5^  (4^+  ")  +  ^^']  =  5§-  ('^^+  ")•  •    •  W 


Subtracting  («)  and  {b)  from  2^3 ,  by  which  the  sine  of  the 
amplitudes  will  not  be  changed  [eq.  (7),  Chap.  Ill],  and  since 
our  new  modulus  is  k^ ,  we  have  for  the  expressions  (a)  and  {d), 


^{6KTu); {a') 


^iAK-u) ip') 

Substituting  these  values  in  eq.  (4),  and  remembering  the 
factor  (i  +  ^03)  introduced  by  each  application  of  eq.  (2)*,  we 
have 

sn  in,  /&)  =  (!+  k:){i  +  k\:)\i  +  k\:f  sn  1^; 


K  K 

sn  ^ {2K ±  «)  sn  ^ (4^±  u) 


sn  ^  (6^  ±  u)  sn  -^  {%K  +  u).         (Mod.  /^,.) 


$6  ELLIPTIC  FUNCTIONS. 

From  this  the  law  governing  the  arguments  is  clear,  and  we 
can  write  for  the  general  equation 

(5)    sn  {u,  /^)  =  (I  +  k:)i^  +  k'Jii  +  k'J  .  .  .  (I  +  k'^-' 

i\.„u         /i.„    .  . 

.  sn  —-Tjr  sn  ——f^  (2A  ±  u) 

K  K 


sn 

2 


^[{2^-2)K±ti 


.sn^{2-K^u).  (Mod..^„.) 

Indicate  the  continued  product  of  the  binomial  factors  by 
A',  and  we  have 

A'  =  {i+ k:){i + k'j{i + k'j{i + ^'„o' . . . 

Since  the  limit  of  k^',  k\^,  etc.,  is  zero,  it  is  evident  that 
these  factors  converge  toward  the  value  unity.  It  can  be  shown 
that  the  functional  factors  also  converge  toward  the  value 
unity.     Thus  the  argument  of  the  last  factor  can  be  written 

From  eq.  (ii),  Chap.  Ill,  we  get  then 

/^      ,     K„U\  ^^2"K  ,^,     ^     r.    , 


But  since  k^  at  its  limit  is  equal  to  unity,  en  =  dn  ;  whence 
the  last  factor  of  eq.  (5)  is  unity. 


DEVELOPMENT  INTO  FACTORS.  57 

From  eq.  (21),  Chap.  IV,  we  have 

Therefore  for  «  =  00 ,  eq.  (5)  becomes 

sn  («,  k)  =  A'  sn  ^^,  sn  ^^  (2Ar±  «) 

•  sn  ^^  (4.^  ±  u)  sn  ^^  (6/i:  ±  7/),     .     .     .     . 


(Mod.  I,) 


or 


7tU  "^        I  7t 

(6)  sn  {u,  k)  =  A'  sn  ^^  [_  IZa  J  sn  ^^  (2/2^  ±  «), 

(Mod.  I,) 

where  the  sign  [IT]  indicates  the  continued  product  in  the  same 
manner  as  '2  indicates  the  continued  sum. 


=  I,  y  F{<t>, 


When  k  =:  I,     I     F(<p,  k)  becomes 


J(      ^<P         ,  ,        I  +  sin  0^ 


J^     cos  q>      -     **    I  —  sm  0 

whence 

I  +  sin  0 
I  —  sin  0' 

and 

sin  0  = ; —  = i =  sn  {v,  i). 


58  ELLIPTIC  FUNCTIONS. 

Hence  in  equation  (6) 


■nu                    iru 

nu 

e      —  e 

sin  -^  = 

hK±u)  = 

httK           -itu                    hvK           nu 

hitK            ■nu                       h-nK            -nu 

q  =  e 

TtK'                        '  nK 

^      a'-e    ^' 

7t 


Put 

(6)* 

and  the  last  expression  becomes 

■nu  nu 

sn  — ^  {2hK  ±u)  =  ^  ^ 


■nu  ■nu    ' 

q       e  -\-q     e 

sn  ^  {2hK-\-  u)  sn  ^^  {2hK  -  u) 

wu  nu  VH  vte 

nu  ■nu  "■«  ■Ku 

g'-'e^'Jrg''e~^'     g' -' e' ^' +  q' \^' 

(■nu  Itu  \ 

^         +^  / 


/    ■nu  ■nu  \ 

q       +^    +\^     +^      / 

From  plane  trigonometry  we  have  the  equations 

: =  —  z  tan  ix,    e"^  -\-  e  "-''  =  2  cos  zx ; 


DEVELOPMENT  INTO  FACTORS,  5^ 


where  i  =.  ^  —  w  which  gives 

nu  .         Ttiu  ,_,   J      . 

^"  '2K'  =  -  ^  *^"  '^' '  (Mod.  i;) 


7t  TC 

sn  —j^i  {2hK-\-  u)  sn  — ^  {2hK  —  u) 

q'-^hj^  g'  ih  _  2  COS  -v^ 
q'-'.hj^q'^h  _|_  2  cos  -^7 

I  —  2q'  ""^  cos  -TF7  +  ^ 
A 

,   ,  ^^^  ,   ,* 

I  +  2q  ^-^  COS  -^-,  -\-  g  *^ 

From  eq.  (10),  Chap.  Ill,  we  have 

sn  {ti,  k)  =  —  i  tn  {m,  k'\ 
Substituting  these  values  in  eq.  (6),  we  have 


tn  {tu,  ^)=A   tan  — ^ 


n 


I  —  2^^*  COS^^/  +^  ■^ 
A 


I  -|-  2/  ^  COS  -T7-,-  -\-  g 


Now  in  place  of  the  series  of  moduli  k',  k^  and  the  cor- 
responding complete  integral  K' ,  we  are  at  liberty  to  substitute 
the  parallel  series  of  moduli  k,  k^  and  the  corresponding  com- 
plete integral  K\  calling  the  new  integral  it,  we  have 

I  —  2^*  cos  -T?  +  ^ 

7tU  K      '     ^ 


(8)  tn  {u,  k)  =  AtSin-^n 


I  +  2^*  cos  -^  -\-  g^ 


K 


6o 


ELLIPTIC  FUNCTIONS. 


=  A  tan 


2K 


TTIt  . 

I  -2/cOS^  +  ^ 

nil 
l-\-2q  cos-^  +  ^ 


I  —  2g  cos  -^-f-q 

717  L 

I +  2/ COS  -j^-\-q' 

7171 
I  —  2^'  COS-^  +  ^ 

7111 
I  +  2^    COS  -j^  +  ^'* 

where 

(9)      ^  =  (I + K){i  +  >^„or(i + ^o3)Xi + -^0.)^  • . . 

Now  ill  equation  (6)  put  u  +  K  for  u,  and  we  have,  since 

cn  2^ 
[eq.  (I  I),  Chap.  Ill]  sn  {u  +  K)  =  ^^, 


cn  «  _  7r(z^  +  -^) 


dn  u 


=  A'  sn 


2K'       L 


n 


7t 


•sn^^[(2/«-  i)iir-«]. 


(Mod.  I.) 


Now  from  2h  —  \  and  2h  +  i  we  have  the  following  series  of 
numbers  respectively : 

2h—i:  I,     3,     5,     7,     9,     etc. 

2/z  +  I  :  3,     5,     7,     9,     etc. 

It  will  be  observed  that  the  factor  outside  of  the  sign  [/7], 

7t{u  -\- K')  ,  ,     .,      ,  ,  ,  1  •  r-m  1 

VIZ.,  sm  am  „-7 — ^,would,  if  placed  under  the  sign  yll  J,  supply 


DEVELOPMENT  INTO  FACTORS.  6 1 

the  missing  first  term  of  the  second  series.    Hence,  placing  this 
factor  within  the  sign,  we  have 

.sn  —i^,\{2h  —  \)K  —  7i\.  (Mod.  i.) 

Comparing  this  with  equation  (7),  we  see  that  the  factors 
herein  differ  from  those  in  equation  (7)  only  in  having  2h  —  i 
in  place  of  2// ;  hence  we  have 


sn  -^,  [{2h  -  i)Ar+  w]  sn  -^,  [(2//  -  i)K -  «] 

(Mod.  I.) 


I  —  2q  ''''-'  COS  -TF-,   +  ^  *'*-" 


I  4-  2/^*-'  COS  -^,-  +  ^'4'4-3 

From  eqs.  (10),  Chap.  Ill,  we  have 
en  {u,  k^  _  I 


dn  («,  k)       dn  (/«,  k')  ' 
whence  eq.  (10)  becomes 


J  I  -  2^'^^-'  COS  -j^r  +  ^'4^-» 

and  when  in  place  of  z«,  /^',  K\  q',  A',  we  substitute  ti,  k,  Ky 
q  3.nd  A,  and  invert  the  equation,  we  have 


(12)         dn  {u,  k)  =  ^  [77] 


I  -\-  2^^-*-'  COS  -r^  -\-  q*^-' 
A. 

I   —  2^^^^-'  COS  -T>-  +  ^*'*~'' 

A 


62 


ELLIPTIC  FUNCTIONS. 


Bearing  in  mind  the  remarkable  property  (Chap.  Ill,  p.  29) 
that  the  functions  sn  ri  and  dn  ii  approach  infinity  for  the  same 
value  of  u,  we  see  that  both  these  functions,  except  as  to  the 
factor  independent  of  ti,  must  have  the  same  denominator. 
Furthermore,  since  sn  u  and  tn  u  disappear  for  the  same  value 
of  u,  they  must,  except  for  the  independent  factor,  have  the 
same  numerator.  Hence,  indicating  by  ^  a  new  quantity, 
dependent  upon  k  but  independent  of  u,  we  have 


71U 


/I  u 

I  —  2q-"  cos  -v>-  -j-  (f 


(13)       sn  («, /^)  =  ^  sin  ^    77 

and  since 

en  «  =  . 

tn  u 

we  also  have,  from  eqs.  (8)  and  (13), 


K 


71 JC 


I  —  2g''''-'  cos  -^  +  ^4/2-2 


sn  u 


<I4)        en  («,  k)  =  -^  cos  ^ 


I  +  ^Q    cos  -^  +  ^ 

r/1 

TtU 
"-      -■  I  —  2g''''-'  cos  -77  +  ^4* -a 


Collecting  these  results,  we  have  the  following  equations 


(15)       sn{u,k)  =  Bsm-^\n\ 


I   —  2^2'^  cos  -^  4-  ^ 
I  —  2^2-^-'  COS   ^  -f  ?"* 


TTM 


(16)        en  («,  z^)  =  ;;5  cos  ^  j^iJj 


I  4"  2^-*  COS  -^  +  ^ 


2q 


'■''—'  COS  VF  +  ^'» 


i^ 


DEVELOPMENT  INTO  FACTORS.  63 

(17)       dn  («,  k)  =  ^  \jl\^ 


I  +  2^^-^-'  COS  -j^  -\-  ^■♦A-a 


I  —  2^^'*-'  COS  -v^  +  ^4/«-2 

To  ascertain  the  values  of  A  and  ^,  we  proceed  as  follows : 
In  eq.  (17)  we  make  ii  =  o,  whence,  by  eq.  (13),  Chap.  II, 
we  have 

whence 

In  equation  (17),  making  u—K,vfQ  get,  by  equation  (i), 
Chap.  Ill, 

2A — I  \  2 


k'--[nY~^    ]=  -• 


(19)  .••  ^  =  ^'. 


We  have  identically 

I 

^  A 

whence 

tiru 

To  calculate   .5,   put   e'     =  v;    if  we    change    —^   into 


64  ELLIPTIC  FUNCTIONS.  ^ 

—r^  -\-  — T3-- ,  y  will  change  into  v  Vg,  and  sn  u  will  become,  by 
eq.  (14),  Chap.  Ill, 

sn  (u  4-  iK')  =  -7 . 

^      '         ^       ksn  u 

7tU  7tU 

Now,  replacing  sin  — r?  and  cos  -^  by  their  exponential  val- 
ues, and  observing  that 

TtU 

I  —  2^"  COS  -^  +  ^''  =  (i  —  q''y''){\  —  q"y~% 

we  have 

B    V  —  V-'       [-iT](i  —  ^V=)(i  —  f'^v-^) 


sn  u  = 


2        ^ITl      [i7](i— (7^*-V^)(I-^^''-V-^)• 


Chang^ng  u  into  «  +  ^^'}  ^^^  consequently  v  into  -y  ^^  we 
have 


/&  sn  «       2  -f/  —  I  [/^](i  -  ^'^^OCi  —  ^*"'0  ' 

Multiplying  these  equations  together,  member  by  member, 
and  observing  that 

—  y  Vq 

V  —  y  =  v{i  —  y% 
we  get 


I   _B*     I  —  qv* 
k  ^  4         y  \/q 


DEVELOPMENT  INTO  FACTORS. 


65 


— (,  -  .OKI  -  oV-;4-^.V.:. 

(I  -  ^v-0(i  -  ^v-Q  .  ■ . 

(i  —  v~-){\  —  ^V"^)  .  . . 


whence 


4 

T 

3  =  2fn 

Substituting  these  values  in  eqs.  (15),  (16),  and  (17),  we 
have 


2  ^ Cl  7tU 

(20)    sn  (u,  k\  =  — rz-  sin  — ^>    11 


nti 


I  —  2^*  COS  >5^  -{-  ^ 


77  7^ 


I  —  2^*"'  COS  -Tr^  +  . 


.4A-2 


TT^^ 


(21)   cn  {u,  k)  = 


m  */-  r-     -1    I  +  26^^^'  COS  7>    -h  ^ 


Vk 


COS 


.^H 


TTZ/ 


I  —  2^''"'  COS  -^   +^ 


mc 


(.)d„(..)  =  v4;7]-- 


I  +  2^"-'  COS  -7^  +  ^ 


K 


2q"'-'  COS  P=^  +  ^" 


CHAPTER   VIII.. 
THE   0  FUNCTION. 

We  will  indicate  the  denominator  in  eq.  (20),  Chap.  VII,  by 
<t>{u),  thus: 

TTU 

(1)  (li{u)  =  [7I](i  -  2^^*-'  cos  ^  +  g^-% 

We  will  now  develop  this  into  a  series  consisting  of  the  cosines 

Tttl  TTU 

of  the  multiples  of  -v?  .     Put  —^  =  x,  whence 
^  K  2K 

2  cos-^  =  (^"  +  ^-="*); 
but 

Tttl 

I  —  2^^*-'  COS  -^  +  q^-^  =  (i  —  ^'*-'^'")(i  —  ^^*-V-^'^% 
and  therefore 

(2)  0(?/)  =  (i  —  ^^"■^)(i  —  ^V^'^)(i  —  /^^'*)  .  . . 

(i  —  ge-^'^'Xi  —  /^-^■*)(i  —  ^V-^"'*) 

Putting  now  u  -\-  2iK'  instead  of  u,  we  have 

_  n{u-\-2iK')  _  nJK' 

X,  -  -j^        -x-\-  -^  , 

27tK' 


^TfX ^    — ^    ^IfX  T^  y 


66 


THE   Q  FUNCTION.  67 

and 


9 


From  these  we  have 


(f){u  +  2iK')  =  —  -  ^-^  (i  —  ge^'){\  —  ^'^)  .  .  . 

(i  —  qe-''%i  —  q'e-'")  .  .  . ; 
whence 

0(«  +  2/Ar')  =  —  -  e-^''  (p{u), 

or 

ir/« 

(3)  0(«  +  2^X0=  -  ^-'  r  "^  0(«). 

Now  put 

TtU  27CU  3^M 

{4)     <p{u)  =  A  -\-  B  cos  T^  +  ^  cos  -^  -{-  D  cos  —^  -\-  etc. 

Since 

cos-^  =K^"  +  ^-""), 

this  becomes 

(5)     (p{ti)  =  A-\-  ^Be^''    +  ICe^'    +  iZ?^^-    +  .  .  . 

+  ^Be-''^  +  iC^-4'"  +  iZ>^-6'^  +  .  .  .  ; 

whence 

I       .    ^,  ,  ABC.         D     . 

(6) e-"^  <^{u)  = e-"' ^^'* e*"  —     .  . 

q  q  2q       2q  2q 

B  C      ^         D      ^. 

£-^** ^~^* ^"^'*  

2q  2q  2q 


68 


ELLIPTIC  FUNCTIONS. 


Now  in  equation  (5)  put  u  +  2iK'  in  place  of  u,  remember- 
ing that  ^-'^  and  ^""'^  are  thereby  changed  respectively  into 
^V"'*  and  q'^'e'"'",  and  we  have 


CI 

2 


2 


2^ 


c: 


'2^*         ' 


2q 


Since  equations  (6)  and  (7)  are  equal,  we  have 

Bz=^-  2gA\ 

C=  +  2q'A; 
D=-2q'A; 


C  _B£ 

2q'~    2  ' 

D  _Cq^ 
2q~    2   ' 


whence 


(8) 


TTU 


[i7](l  -  2^^*-'  COS  -—-f  ^^-) 


TT?/ 


=  yi(l  —  2q  cos  -r^  +  2q*  cos  — ^  —  2/  cos  ^-^ 


4-  2^'    cos  -^-  '  ■  •)• 


The  series  in  the  second  member  has  been  designated  by 
Jacobi  and  subsequent  writers  by  ©(?/),  thus: 


(9) 


nu 


27Itl 


&{u)  ■=  I  —  2q  cos  -^  -f"  2^^  cos  — ^   — 


CHAPTER   IX. 
THE   0  AND  H  FUNCTIONS. 
In  equation  (20),  Chap.  VII,  viz., 


UTt 


1  '^^      ,  u 

I  —  2^^*"'  COS  T^  -|-  q^' 


the  numerator  and  the  denominator  have  been  considered  sej>- 
arately  by  Jacobi,  who  gave  them  a  special  notation  and  de- 
veloped from  them  a  theory  second  only  in  importance  to  the 
elliptic  functions  themselves. 

Put  [see  equation  (8),  Chap.  VIII] 

(1)  Q{u)  =  ^  [77](i  -  2^*-  cos  ^  +  ^-). 

(2)  H{u)  =2-  ^^  sm  ^  |^77j(i  -  2$r^*  cos  ^  +  ^) ; 


A   being  a   constant  whose  value  is  to  be  determined   later. 
From  these  we  have 

(3)  sn  (u,  k)  =  — r  .  ;4-T  . 

69 


•JO 


ELLIPTIC  FUNCTIONS. 


The  functions  sn  u  and  en  ii  can  also  be  expressed  in  terms 
of  the  new  functions  ;  thus  we  have 


7tU 


fk'  nuV    n^+^^'*    cos-^+^ 

(4)     en  {u,  k)  —  \         2  ^~q  "" 


cos 


M 


2K\—    _l  '^it 

1—2(7"''-'  cos  -T^  +^'"' 


or,  since 


sin  x^^  cos  \x  -(-  -j  and   cosjr  =  —  cos  f;tr-|-— j, 


2Kx 
and  putting  tc  =  ~z~i 


en 


l2Kx 


?.^) 


.>-[-^(^+f 


=x/^ 


p-r-f^+-] 


K^i 


Replacing by  its  value,  u,  we  nave 


(5)  en  («,  k)  = 

.   Furthermore, 


+^) 


©(«)   * 


(6) 


dn  («,  /&)  =  i^  \ll\ 


I  +  2^*-'  cos  -^  +  ^^-^ 
I  —  2^*"'  COS  -vr  +  ^"* 


THE   Q  AND  H  FUNCTIONS.  Jl 

gives  in  the  same  manner  • 


,    2Kx         /-^ 
dn =  Vk' 


j^j^\ 


\     7t     I 


or 


(7)  dn  («,-&)  =  1/1'^^. 
If  we  put 

(8)  H{u^K)^HSti\ 

(9)  B{u^K)^Qiu), 

the  three  elliptic  functions  can  be  expressed  by  the  following 
formulas : 

(11)  ^"(^^''^)  =  V^'0(^' 

(12)  dn(«,>&)  =  i/F|^g. 

These  functions   0  and  U  can  be  expressed  in  terms  of 
each  other.     By  definition, 

Hiii)  =  2CVq  sin  ^'i_    Ji^  "  2^ cos  ^  +$^j  ; 


72  ELLIPTIC  FUNCTIONS. 

but 

TtU  f  '""  ^~'\  /  ffM  V-i\ 

I  —  2/  COS  -^  -|-  <7-''  =  {  I   —  $rV      A-      1  (  I   —  ^*^  :^~ j, 


Tk         ~  sic 
TtU       e       —  e 


iriu 


—niu  j^ 


and  consequently 

ir»«_  /  jria\  /  iriu  \  /  jr/a\ 

(13)  H{u)=Cy/ge''~'^  V~i\i-e^)[i  ~  g^'"""  )\i-q^e^). . . 
Now,   changing  u  into   u  -\-  iK',  and    remembering    that 

-ttA'' 

e  ^  :=  g,  we  have 

(14)  H{u  +  iK') 

—iriu  I  viu\  f  jr?VA  /  viuX  I  7riu\ 

=  Cg~^c^V~i[i-ge^][i-ge'^l[i-g^e^l\i-g^e~'^l...; 
and  reuniting  the  factors  two  by  two,  this  becomes 

(15)  H{u  +  zK^) 

_^/  TTU      ,         \(  ,  TTU  \ 

=  CV-iq-^e   ^k\i  -2^  cos  -j^^g^j\i-2g'cos^-{-g'j...; 
and  finally,  according  to  equation  (i). 


(16)  H{u  +  iK')  =  V^^^g  V   '''@{u). 


THE   Q  AND  H  FUNCTIONS.  73 

In  the  same  manner,  we  can  get 

(17)  ^{u  +  iK')  =  V^~if'e~  ^H{u). 
Substituting  u  +  2K  for  u  in  equations  (i)  and  (2),  we  get 

(18)  e{u -]- 2K)  =  e{u), 

(19)  H{u -{- 2K)  =  -  H{u), 

7t  71U  It  TtU 

since  cos  j^{u  -\-  2K)  =  cos  -j^ and  sin  — ^(;^-|~2^)=  — sin  — ^. 

The  comparison  of  these  four  equations  with  equations 
(10),  (11),  and  (12)  shows  the  periodicity  of  the  elHptic  func- 
tions. For  example,  comparing  eqs.  (10)  and  (16)  and  (17),  we 
see  that  changing  u  into  u  -\-  iK'  simply  multiplies  the  nu- 
merator and  denominator  of  the  second  member  of  eq.  (10)  by 
the  same  number,  and  does  not  change  their  ratio. 

The  addition  of  2K  changes  the  sign  of  the  function,  but 
not  its  value. 

We  will  define  0,  and  H,  as  follows : 

(20)  &lx)=Q{x-\.K)', 

(21)  H^x)  =  H{x -^^  K). 
Hence  we  get,  from  equation  (17), 

Q^x  -f  iK')  =  e{x  +  iK  J^K)  =  Q{x^K-\-  iK) 

=  iH{x-{-K)e   ''^ 

=  iHlx)e  '"^  (-V^i). 

irr 
~   T  7t  / .       7C  > 

smce  e      =  cos r  —  i  sm  -    —  —  V  —  i  ; 


74  ELLIPTIC  FUNCTIONS. 

whence 

(22)  Blx-\-iK')=Hi^x)e   '"^ 
In  a  similar  manner  we  get 

(22)*  ^,(-^+^XO  =  0,(^)^    '^'^ 

2  A^™ 

In  eq.  (9),  Chap.  VIII,  put  u  =  — -,  and  we  get 

71 

,     ,         j2Kz\ 

(23)  ^l"^  j  =   1—2^  COS  2^^  +  2$''  COS  43'  —   .  .  . 

Now,  in  this  equation,  changing  z  into  ^^  -) —  ,  and  observing- 
eq.  (20),  we  get 

,     ,        ^,  l2Kz\ 

(24)  e), ! 1  =  \  ^2q  cos  2z  -{-  2^'  cos  4^  -{-  .  .  . 

Applying  eq.  (22)  to  this,  we  have 

=  ,.-y[l+2^COS2(^+^')+2^^COS4(^  +  ^-)+...] 

f 


THE   Q  AND  H  FUNCTIONS.  75 

=  ^'V  [I  +  q'e'''  +  ^"^'"'^  +  .  .  . 

=  ^*  [^"  +  /^'"  +  ^"^'"  +  •  •  • 

=  2^»  [cos  z-\-q^  cos  S^'  +  q"  cos  S^"  +  .  .  •]  ; 
whence 

(25)  ^,  (^)  =  2  t^  cos  5:  +  2  1^^  cos  3^  +  2  v'^  ^'  cos  5^  +  .  .  . 

In  this  equation,  changing  z  into  ^  —  - ,  and  applying  eq. 
(21),  we  get 

(26)  H  f=-^)  =  2  t^^sin  ^  —  2  1^^  sin  3-3^+2  ^'f'  3in  5^  -  . . . , 


smce 


-.(^1=-(^^+4 


We  will  now  determine  the  constant  A  of  eq.  (8),  Chap. 
VIII,  and  eqs.  (i)  and  (2)  of  this  chapter.  Denote  A  by/(^), 
and  we  have 


TIU 


(26)*         [iT](i  -  2^^*-'  cos  ^  +  ^"0  =  Aq)^^u). 
Substituting  herein  ?<  =  o  and  «  =  — ,  we  have 

[7i](i  +  ^-)=y(^)0(f). 


76  ELLIPTIC  FUNCTIONS. 

From  eq.  (9),  Chap.  VIII,  we  get 

(27)  0(o)  =  I  -  2^  +  2^*  -  2/  +  2^"  -  .  .  . ; 

(28)  ©(f)  =  I  -  2^^  +  2^-  -  2^-  +  2/^  -  .  .  . ; 

from  which  we  see  that  0(o)  is  changed  into  ©( — j  when  we 

put  q'  in  place  of  q. 
Whence 


[77J(l_^-4y=/(^^)0(f); 


and  therefore 


(29) 


,7l-2 

(I  -q""^' 


I 


/"-")( I — ^''"0 

Now,  the  expressions  4//  —  2,   8//  —  4,    and    8/^    give   the 
following  series  of  numbers  : 

4/1  —  2,  2,     6,     10,     14,     18,     22,     26,     30,     34  ; 

U  —  4,  4»  12,  20,  28,  36 ; 

SA,  8,  16,  24,  32. 

Hence,  the  three  expressions  taken  together  contain  all  the 
even  numbers,  and 

\_n]{i  -  q^-'){i  -  g^-%^  -  f)  =  \jt\{i  -  't\ 

Therefore,  multiplying  eq.  (29)  by 


H 


we  have 


n 


I  —  q^ 


JI  -q- 


THE   0  AND  H  FUNCTIONS.  J  J 

Now  in  this  equation,  by  successive  substitutions  of  q*  for  q^ 
we  get 


fW) 


n 


Ji  -  ^' 


/(^.Trrli-^' 


=  Mf^ 


,128* 


f(qn  ~  L    J  I  -  q'^'"  ' 


Now  q  being  less  than  i,  q"  tends  towards  the  Hmit  o  as  « 
increases,  and  consequently  i  —  q"  tends  towards  the  limit  I. 
Also,  from  eq.  (8),  Chap.  VIII,  we  see  that  /{o)  =  i.  Hence, 
multiplying  the  above  equations  together  member  by  member, 
we  have 

(30)  m=[n]^, 

or 

(31)  A  = 


{l-q^){l-q*){l-q^) 


Substituting  this  value  in  equation  (8),  Chap.  VIII,  we  have, 
after  making  tt  =  o, 

n       .Vrr        ^'Wt        .^V  I  -2q  +  2q*-2q' -}-... 

m 

~(i-?^)(i-?o(i -/)...' 

(See  equation  (9),  Chap.  VIII.) 

Transposing  one  of   the  series  of   products  from  the  left- 
hand  member,  we  get 

f    _    \/    _    t\ §(P) 

(I        q^l        q) (I  _  ^)(i  _  ,/)(!  _  ^^)(i  _  ^^)  .  .  .• 


78 


ELLIPTIC  FUNCTIONS. 


Introducing  on  both  sides  of  the  equation  the  factors  i  —  ^% 
I  —  ^\  I  —  q^,  etc.,  we  get 

(I  -  q\l  -  ?')(!  -  ?')(!  -?')... 

=e(o)i^/.i^.i^:.i^:... 

^  ''    I  —  q      i  —  q      I  —  q      I  —  q 

=:0(O)(I+^)(I+^=)(I  +  ^O--.; 
whence 

<32)         ©(o)  =  (,+y)(,+/X.+?T 

Resuming   equation    (20),   Chap.    VII,  and    dividing   both 
members  of  the  equation  by  u,  we  have 


7tn  ,  mi.         . 

,,-     sm  — r>  I  —  2^     cos  -T?r  -\-q* 

snu       2Vq  2K1-    -  ^  /r    '   ^ 


l/>^ 


A' 


I  —  2q-"~'  cos-^  +^ 


This,  for  ?/!  =  O,  since  the  limiting  value  of for  Zi(  =  o  is  i, 

7r7( 
sm  -7> 

and  of   for  ;ir  =  O  is  — t-_,  becomes 

u  2K 


^q      n     (I  -^7(1-^7(1-/)'... 
-^/J'-^K'    {l-qni-qJil-qy..: 


or 


(33) 


VkK 


■{l-qyi-qy,-q^)...-V^ 


(34) 


L(l  -  ^)(l  -q'){l  -f) 
Further,  from  equation  (21),  Chap.  VII,  for  u  =  O,  we  have 

Vk  V{l+q%l+q'){l+q'). 


TV  Vq 


2\'J'  'Vq        L(I  -^)('  -/)(!-/)  •••J 


THE   0  AND  H  FUNCTIONS. 
The  quotient  of  these  two  equations  gives 


79 


<35) 


2Vk'K 


or,  substituting  the  value  of   \' k'  from  eqs.  (i8)  and  (19),  Chap. 
VII, 


-(i+^)(i+^o(i+^o..  J* 


2k' K    r(i 

Comparing  this  with  equation  (32),  we  easily  get 
<37)  ®(o) 


2k'K 


From  equation  (9),  Chap.  VIII,  making  u  =  K,  we  get 

<38)  Q{K)  =  I  +2^+2^^  +  2/  +  2/«+  .  .  . 

Making  ^r  =:  o  in  equation  (24),  Chap.  IX,  we  have 

(39)  ©,(0)  =  I+2^+2^'  +  2/+... 

This  might  also  have  been  derived  from  eq.  (38)  by  observ 
ing  that 

Knowing  &{o),  it  is  easy  to  deduce  Q{K)  and  H{K). 
From  equation  (7)  we  have 

dn  «  =  Vk'  — pv^- 
Making  «  =  o,  we  have,  since  dn  (o)  =  i, 


(40) 


-(->  =  ?|- 


80  ELLIPTIC  FUNCTIONS. 

From  equation  (5)  we  get,  in  the  same  manner, 

(41)  H{K)  =  y^l  r.(o). 

From  eq.  (12),  Chap.  IX,  we  have 
(41)*  dn  u  =  X^'i  -I^  sin-^  0  =  1/F  ^i^  ; 

nu 
and  putting  x  =  —f^,  we  have 

dn  ?/       \  -\-  2q  cos  2x  -\-  2q*  cos  4;ir  -j-  2q^  cos  6x  -\-  .  .  . 
^4"/      ^  rr       I  —  2^  cos  2x  -\-  2g*  cos  4^1;  —  2^"  cos  6x  -\-  .  .  .' 

Putting 

dn  u 
(42)*  -;^  =  cot  r, 


we 


have 


cotv— I            -     o       .          cos2;f4-/(4cos'2:i;— 3  cos2;r)-}-... 
c-5FH^='='"(45  -r)=21 :  +  ^X4cos-2^-2) = 

whence 

tan  (45°  -  :k)[i  +  q\A  cos'  2;ir  -  2)] 

(43)  cos  2x  = — 

—  ^"(4  cos'  2X  —  3  COS  2x)  —   .  .  .  , 

and  approximately, 

/     ^  o         tan  (45°  -  r) 

(44)  cos  2X  =  ~ -. 

From  equations  (37)  and  (40),  Chap.  IX,  we  have 


X  = 


(45)  -  -  0=(a:)  ' 

whence 

(46)  u  =  xQ\K). 


CHAPTER   X. 
ELLIPTIC   INTEGRALS   OF  THE  SECOND  ORDER. 
From  Chap.  I,  equation  (19),  we  have 

E{k,  <f))  =y*  Vi  -  k'  sin^  0  .  d(f)  =r^<t>  .  ^0. 
From  this  we  have 

£(0)  +  £(^)  =y*^0  .  d(t>  +r^<l>  .  d(l>. 

Put 

(i)  £<P  +  £^p  =  S. 

Differentiating,  we  get 

(2)  ^4>.d(p-\-/}tp  .dip=  dS. 
But  we  have,  Chap.  II,  equation  (2), 

or 

(3)  Atp  .  d(p  -{-  Act> .  dip  =  o. 

Adding  equations  (2)  and  (3),  we  get 

(4)  (//0  +  ^>P){dcp  +  dip)  =  dS. 

8x 


82  ELLIPTIC  FUNCTIONS. 

Substituting  cos  >u  from  eq,  (5),  in  eq.  (5)*,  Chap.  II,  we  get 
sin  0  cos  ipA}x  -j-  cos  0  sin  rp 


(5) 


whence 


J0  = 


sin  }x 

sin  tp  cos  0^yu  -\-  cos  ^  sin  0 
sin  /f 


(6)  J0  ±  J^  =  ^=^  sin  (0  ±  ^). 

Substituting  in  equation  (4),  we  have 


(7) 


^5  =     '".    '    '  sin  (0  +  ^•)40  +  ^) 
sin  /^  '    '  /  \      1    '  / 

^y"+  I 


sin  }x 
Integrating  equation  (7),  we  have 


«f  cos  (0  -f-  ^). 


E(p^Eip  = 


sin  yU 


[C-cos(0  +  V)]. 


The  constant  of  integration,  C,  is  determined  by  making 
0  =  0;  in  this  case  f  =  M^  ^4>  =  o,  -£^  =  £m,  and  5  =  ^yw ; 
whence 

Ja^4-  I 


^M  =  —^. — ■ — -  (C  —  cos  /Af 

and  by  subtraction, 

^u  4-  I 
£^  -^Eip  —  Em  =  — ^ —  (cos  /I  —  cos  0  cos  ^  +  sin  0  sin  ip). 

But,  Chap.  II,  eq.  (5), 

cos  jj.  —  cos  0  cos  ^  =  —  sin  0  sin  tp^/a ; 


ELLIPTIC  INTEGRALS  OF    THE    SECOND   ORDER.  83 

whence 

I  —  A*  IX 
E(p  -\-  Eri)  —  Eu  =  — ^ sin  0  sin  ^ 

sin  yW  f  r 

whence 

(8)  E(p  -f-  Eip  =  E/x  -\-  J^  sin  0  sin  ^  sin  /i. 

When  0  =  ^,  we  have 

(9)  E}x  =  2E<t>  —  k^  sin'  0  sin  //. 
But  in  that  case 

(10)  cos  //  =  cos'  0  —  sin'  0J/i ; 
whence 


(11)  sin  0 


/l    —  cos  }A 


Let  0,  01 ,  0j ,  etc.,  be  such  values  as  will  satisfy  the  equa- 
tions 

(12)  £0  ==  2E(f)^  —  k"  sin'  0^  sin  0, 

.£0^  =  2£"0j  —  k"  sin"  0j  sin  0j  , 

etc.  etc. 

Assume  an  auxiliary  angle  y,  such  that 

{13)  sin  ;^  =  >^  sin  0; 

whence 

2^0  =  cos  y^, 

and  Chap.  IV,  eq.  (24), 

f    \  .  sin  i0 

<I4)  sin  01  = ^. 

^       cos  1^;^ 


84  ELLIPTIC  FUNCTIONS. 

Applying  eqs.  (13)  and  (14)  successively,  we  get 


sin  -J0 
sin  01  = T- ,     sin    v§   =  /&  sin  0i ; 


(15) 


sin  -101         .  ,    . 

sin  0i  = r—  ,     sin  T/i  =  >e  sin  0i  : 

*       cos  \Yi^  '  ^  ^ 

sin  ^0  I 


sin  0 1  = 


—      cos  iv  I    ' 
whence 
(16)     ^0  =  2*^0^^  —  (sin  0  sin'  y^  -\- 2  sin  0^   sin'  ^/-j 

-f  2"  sin  0j   sin'  ^j  -|-  •  •  •  2""'  sin  0j_  sin'  y    i 

To  find  the  limiting  value,  ^0i  ,  we  have,  by  the  Binomial 
Theorem,  since  sm0=i  —  p-j-p etc., 

J0  =  (i  ~  y&'  sin'  0)* 


=  ^-2-^'+V6        8. 


(l-f)^- 


Whence 


£y^0,  =    /"'"  J0.^0 


2"  t/o 


(17) 


'4:        6 


/&'(4  -  3^^ 


=  0,  -^0S+'^"_,"— ^0S. 


120 


ELLIPTIC  INTEGRALS  OF    THE   SECOND   ORDER.  85 

Substituting  in  eq.  (16)  the  numerical  values  derived  from 
equations  (15)  and  (17),  we  are  enabled  to  determine  the  value 
oi  E<t>. 

Landen's  Transformation  can  also  be  applied  to  Elliptic 
Integrals  of  this  class. 

From  eq.  (11),  Chap.  IV,  we  get,  by  easy  transformation, 

(i8)  sin"  20  =  sin*  0,  (i  -f-  >^,  +  2^.  cos  20). 

From  this  we  easily  get 

2^0  cos  20  sin'  0,  =  sin"  20  —  sin*  0,  —  k^  sin'  0, 

=  I  —  cos'  20  —  sin'  0,  —  k^  sin'  0, 
=  A*k^<p^  —  sin'  0j  —  cos'  20 ; 
whence 

cos'  20  -|-  2k^  sin'  0,  cos  20  =  -^'^o^i  —  sin'  0, ; 
and  from  this. 


cos  20  =  —k^  sin'  0,  ±  ^A'*k^<t>^  —  sin'  0,  +  k^  sin*  0, 

(19)  =  cos  0,  J>^.0,  -  >^.  sin'  0, ; 
whence,  also, 

I— cos'  20  =  I— cos'  0,^'0,-|-2i^  sin'  0, cos0,J>^„0,— ^/  sin*  0, 

=  sin'  0,  (i -f^.'  cos'  0i4-2>&,  cos  0,  Ak^<t>x—K  sin'  0,) 
and 

(20)  sin  20  =  sin  0,(J^„0,  -|-  >^^  cos  0,). 
Differentiating  equation  (19),  we  get 

^0         .     .  (>^o  cos  0,  +  ^/&„0,y 
2  sin  2<p-^  =  sin  0.  ^-^^ )^ ?^^. 

^0,  ^/^o0j 

Dividing  this  by  equation  (20),  we  have 

2d^_  k^  cos  0.  -f-  J/^„0, 
^0.  ~  Ak^<P^ 


86  ELLIPTIC  FUNCTIONS. 

But  from  (19),  and  eq.  (6),  Chap.  IV, 

z.^    •  »  ^       ^'(i- cos  20) 

k  sin   0  = 

2 

2k 
=  (i    I  °^y  1 1  +  ^0  sin"  0,  -  cos  0,  /J/^„0, \ 

whence 


and 


and 


AU^  ^40.  +  ^,  cos  0^ 

Ak<t>  = ^^^ , 

oAhrh    ^         {K  COS  0.  +  AkM 
^^^*^V0.  -  (I  +  /^„)Z/X'„0,        ' 

dcf),      {k,  cos  0,  +  Ak,<l)y 


dcf)Ak<p 


This  gives  immediately,  by  integration, 

I       r  d(p. 


2(1 +i„)/   z?/^„0, 


1  2^=^00.  +  2>^„  COS  0,^40,  -k:'\ 


Thus  the  value  of  Ek(p  is  made  to  depend  upon  Ek^(p^ 
(containing  a  smaller  modulus  and  a  larger  amplitude),  and 
upon  the  integral  of  the  first  class,  /^^„0, ;  k„,  cf>^,  etc.,  being 
determined  by  the  formulae  (6)  to  (12)  of  Chap.  IV. 

By  successive  applications  of  equation  (21),  Ek(t>  may  be 
made  to  depend  ultimately  upon  Ek^„<p„,  where  k^„  approxi- 
mates to  zero  and  Ek^„cp„  to  0„ . 

Or,  by  reversing,  it  may  be  made  to  depend  upon  Ekn<p^n> 
where  k^  approximates  to  unity  and  Ek„(p^„  to  —  cos  0„„. 


ELLIPTIC  INTEGRALS  OF   THE   SECOND   ORDER.  87 

To  facilitate  this,  assume 

Gk<p  =  Ek(i)  —  Fk4>. 
Subtracting  from  equation  (21)  the  equation 

Fk<t>  =  l±^Fk,<p,  (see  eq.  (13),  Chap.  IV), 
we  have 

^^^  =  J   .  ^  (^^o<^i  +  >^o  sin  0j  —  k,Fk^(t>^, 
Repeated  applications  of  this  give 

GK<t>,  =      .,    (g>^.,0»  +  /^oo  sin  0,  -  k,,Fk,,(p,\ 


6^/^„(«-,)0«-i  =  T  -1-  h   (^^on0n  +  4«  sin  0«  -  k,„F^,„<p„). 
Whence 
(22)         C^0  =  2.'  i  ^>7  ^-  -  J'^'-f')  I  +  _^^..'>. 


[7ri(i+/f..)  [n](i  +  ,f..) 

n  n 

But  since  (compare  eq.  (13),  Chap.  IV) 

Fk,„cp„[n]{i+k,:) 


Fk(p  = -, 

2' 

or 

Fk„„<P„  2^Fkct> 


(23) 


[7I](i+/^„„)       [7T](i +/&„„)' 


88 


ELLIPTIC  FUNCTIONS. 


and  since,  also,  (compare  eq.  (6),  Chap.  IV,) 


o(«-i) 


we  have 


(24) 


2«/^„ 


[77]  (1  +  ^0  J 


k 

2" 


k 


n 


o(«-i) 


l_   M   _J  'i-Q^t 


n 


f^i>{n-i)    \     T 


n 


2"    '  k'  '  k 


^    n 


'^o(m-i) 


00  '^an  L.  M  _l 


^o[n-i)  • 


Substituting  these  values  in  equation  (22),  and  neglect- 
ing the  term  containing  Gk^n(p„  since,  carried  to  its  limiting 
value, 

=  0„  —  0„  =  o,  {n  =  limiting  value,) 

we  have 

f   k  VJ^„  sin  <p„  ifl]  ^^o{n-z)  —  k^  \ir\  ^o(«-i)   ] 

(25)    Gkc}>  =  2:  -{  


2» 


=  k 


L_     2 

k'  r 


-^  sm  0,  +  --^r-  sm  0,  H -, sm  03  -f  .  .  . J 


2 
whence 


I  +  2  +  ^T-  +  — -i—  +  .  .  J; 


(26)     £/&0  =  /'y^0 


fG+t-+^#+---)] 


+  '^L~^sm0,H ^^sm0,H -, sm  0^  +  .  .  .  J  . 


ELLIPTIC  INTEGRALS  OF   THE   SECOND   ORDER.  89 

From  eq.  (3),  Chap.  V,  we  see  that  when  0  =  -, 
0„  =  2''"'7r. 

Substituting  these  values  in  equation  (26),  we  have  for  a 
complete  Elliptic  Integral  of  the  second  class, 

A''  ?)[■  -  f-G +I-" +^"+%'^+  •  •  •)]■ 

In  a  similar  manner  we  could  have  found  the  formula  for 
E{k,  <p>)  in  terms  of  an  increasing  modulus,  viz., 

<28)    E{k,  </>)  =  F{k,  0)  [i  +  ,&(i  +  I  +  |l^  +  ^^+  .  .  . 
2"-"  2"-'         \~| 

fS^fS^     .     .     .     K„_2  ^1*2     •     •     •     f^n-J-1 

-k[_sm  0  +  :^  sin  0.  +  :;^-  sin  0.+  .  .  . 

2»-i  2"  ~\ 

\kk^  .  .  .  -^,_,  Vkk,  .  .  .  >&..,  -I 


V'     w-<-  i--  i''i<- 


CHAPTER   XI. 
ELLIPTIC  INTEGRALS  OF  THE  THIRD  ORDER, 
The  Elliptic  Integral  of  the  third  order  is 

(1)  n{n,k,^)=J^    (i+^sin'0)J0- 

Put 

(2)  77(0)  +  77W  =  S; 

whence  we  have  immediately 

,.  r^ ^__ , ^^ 

K3)  ^•^-  {i  -I-  n  sin=  0)J0"^  (i  +  n  sin'  tp)^f 

But,  eq.  (2),  Chap.  II, 

d(p      dtp 

whence 

^^^  f  I L__i^ 

\I  -|-  ^  sin"  <p       I  -\-  n  sin"  ipi  A(p 

«(sin"  'Z-'  —  sin"  0)  <af0 


(5) 


(i  +  ^  sin"  0)(i  +  «  sin"  ^)     J0* 


From  equation  (8),  Chap.  X,  we  get  by  differentiation,  since 
cr  (or  //)  is  constant, 

Acf)  .  d(p  -\-  Alp  ,  d<p  —  k^  sin  a  ^(sin  0  sin  ^), 

or,  from  equation  (3), 

d(t> 
(sin"  ^  —  sin"  0)  -r-.  =  sin  o-^(sin  0  sin  ^). 

90 

4 


ELLIPTIC  INTEGRALS  OF   THE    THIRD   ORDER.  Ql 

This,  introduced  into  equation  (5),  gives 

n  sin  adisva.  0  sin  0) 
^  \-{-  n  (sin'  0  -|-  sin'  ^)  -j-  «"  sin"  0  sin'  ^* 

Put 

sin  0  sin  ^  =  ^,    sin'  0  -j-  sin'  rp  ■=  p\ 
whence 

/AX  ^c  ^  sin  (Tdq 

^^)  ^'^=  !+«/+«'/• 

From  equation  (5),  Chap.  II,  we  have 

cos  (T  =■  cos  0  cos  fp  —  sin  0  sin  ipJa; 

from  which  we  easily  get 

(cos  cr  +  ^^(^y  =  cos'  0  cos'  fp) 

=  (i  —  sin'  0)(i  —  sin'  tp) 

and  thence 

/  =  I  -|-  ^^  —  (cos  o-  -j-  ^Jcr)' 

=  sin'  cr  —  2  cos  crJo-^-(-  ^'  sin'  <r  .  5^. 

This,  substituted  in  eq.  (6),  gives 

n  sin  (Tdq 


dS  = 


I  -^  n  sin'  (T  —  2n  cos  aAaq  -\-  n{n  -j-  >^'  sin'  cr)^ 
«  sin  crt?'^ 


"  ^  -  2^^  +  C^  ' 

where 

A=  I  -\-n  sin'  c, 

B  =^  n  cos  cr^o", 
^  =  «^'  sin'  (T  -j-  «'. 


92  ELLIPTIC  FUNCTIONS. 

From  this  we  get 

S  =  n  ^vci  (7   I  1 L  Const. 

J  A  -2Bg-\-  Cq'  ^ 

In  order  to  determine  the  constant  of  integration  we  must 
observe  that  for  (p  =z  o,  ij)  —  a  and  q  —  O',  whence 

T1(T  =  «  sin  o-  /     . ^ L  Const  • 

whence 


.S  =  Her  -\-  n  sin 


'J:^ 


dq 


2Bq^Cq'' 
or 

<7)    ncl>-{-m=na+nsmcT  T- f^   ,    ^  ,. 

J,    A-  2Bq  +  Cq' 

But  we  have 


AC~B'  +  {Cq-BY 
CM  dq 


AC-B'  /    Cq-B    V 

^~^\\/AC-B\ 

Cdq 


M  VAC-B' 


VAC-B'       j    , 


/    Cq-B  y 
KVAC-R} 


where  J/  =  «  sin  c. 

The  integral  of  the  second  member  is 

M  Cq-B 

fan -I ? • 


tan 


VAC-B"  VAC-B"' 


ELLIPTIC  INTEGRALS  OF   THE    THIRD   ORDER.  93 

whence 


/' 


',^      o  M        V  Cq-B      ,  B        -^ 

dS=S,  =  —==   tan-'  —^ =  +  tan-' = 

'       VAC-B'^  VAC-B'^  VAC-B'A 


or,  since 

x4-y 
tan"' ;r +  tan-' r  =  tan'' '■ , 

5,  =  — ^—  tan-^jgg^. 
'       VAC-B'  A-Bq 

Substituting  the  values  oi  A,  B,  C  and  M,  we  have 

AC  -  B"  =  n{i  -^  n  -  zJV)(i  +«  sin'  a)  -  n'  cos'  aA'cr 
=  «(i  -[- 11  —  ^'(T  -\-  n{i  +  ^)  sin'  (T  —  «JV) 
=  «(i  -|-  «)(i  —  JV  -[-  n  sin*  <t) 
=  «(l  -|-  «)(/^'  -|-  n)  sin'  (T ; 
and  putting 

n  ' 

we  have 

VAC—  B"  —nVhsm  a. 
Substituting  these  values  in  eq.  (7),  we  have 

17(«,  k,  (p)  +  n{n,  k,  tp)  -  n{n,  k,  a)  =  S, 

I  n  Vn  sin  0  sin  ip  sin  cr 

=  — =  tan- 


y  ^2  I  +  ^  ^^^^  (^  —  «  sin  0  sin  ^  cos  (r/d(r' 


CHAPTER   XII. 

NUMERICAL   CALCULATIONS,    q. 

CALCULATION  OF  THE  VALUE  OF  q. 
From  eq.  (7),  Chap.  IX,  we  have 

-whence,  eq.  (9),  Chap.  IV,  eqs.  (27)  and  (39),  Chap.  IX, 


1  +  2^  +  2/  +  2q'  +  2^'"  +  .  .  . 
=  I  -  4^  +  8/  -  16/  +  32^*  -  56/  + .  .  . 
The  first  five  terms  of  this  series  can  be  represented  by 

1  +  2^ 

From  this  we  get 


1  I  —  l^cos  d 

2  I  -|-  r  COS  c' 

which  is  exact  up  to  the  term  containing  q'. 

Or  we  can  deduce  a  more  exact  formula  as  follows :     From 
eq.  (I), 


I  4-  j/cos  B  _   \^i-\-  tan'  \B  -f  \/\  -  t"an'  j^ 
I  -  Vcos  6  ~   Vi  +tan'^  -  Vi  —  tan=  ^0 

^     1+2^  +  2^-+...  ^ 

2^  +  2/  +  2^"  +  .  .  .  • 

94 


NUMERICAL    CALCULATIONS,     q.  95 

whence,  by  the  method  of  indeterminate  coefficients, 

(3)^  =  itan=^  +  ,Vtani^|+AVtan'°|+^|^tan-^^+..., 

or 

d 
\o^q  —  2  log  tan  -  -  log  4  + 

log(i  +itan^|  +3^  tan'l  +  AVtan"f  •  •  •) 
<4)       =  2  log  tan  -  -  log  4  4- 

M{\  tan^  \  +  ii^  tan'  |  +  /^  tan"  j  +  .  •  .)> 

J/  being  the  modulus  of  the  common  system  of  logarithms. 
Put 

e  0 

<5)         log  ^  =  2  log  tan  -  +  9-397940  +  a  tan*  -  + 

a  a 

^tan'-^-ttan"-  +.  .., 

in  which 

log  «  =  9.0357243; 

log  b  =  8.64452  ; 

log  c  =  8.415 18  ; 

log  ^=  8.25283. 

Example.     Let  k'  =  cos  10°  23'  46".     To  find  q. 

a  n 

4  log  tan  -  =  5.835  2  log  tan  -  =  7.9176842 

log  a  =  9.036  9- 3979400 

74 


4.871 


log^=  7.3 1 563 16 
a  tan*  -  =  0.0000074 


96  ELLIPTIC  FUNCTIONS. 

e 

When  6  approaches  90°,  tan  —  differs  little  from  unity,  and 

the  series  in  eq.  (5)  is  not  very  converging,  but  g  can  be  cal- 
culated by  means  of  eq.  (6),  Chap.  VII,  viz., 

K  t  K' 

q  =  e         ,  q  =  e       . 

By  comparing  these  equations  with  eqs.  (6)  and  (9),  Chap. 
IV,  we  see  that  if 

g=f{k)    =/(^. 

then 

Therefore,  having  6,  we  can  from  its  complement,  90°  —  6^ 
find  g'  by  eq.  (5),  and  thence  q  by  the  following  process.  We 
have 

T  irA"  T  vfC 


whence 


-   =   ,^,  i,=   .A-; 


log  -  log  -  =  Wn*  =  1. 86 1 5228, 


(6)  log  log  -  4-  log  log  -,  =  0.2698684, 

by  which  we  can  deduce  q  from  q'. 

Example.     Let  6  =  79°  36'  14".     To  find  q. 

90°  -  6^  =  10°  23'  46". 
By  eq.  (5)  we  get 

log  ^  =  7.3 1  563 16,  log  -,  =r  2.6843684, 

and     log  log  -  =  .4288421  ; 


NUMBRJCAL  CALCULATIONS,    f.  97 

and  by  eq.  (6), 

log  Jog  -  =  9-8410263 ; 
whence 

log  y  =  1.3065321. 

When  i'  =  it  =  COS  45"  =  i  ♦2,  eq.  (6)  becomes 

(7)  l<^i  =J/)r=  1.3643763;  (it  =  if:) 

whence 

1<^  ^  =  2.6356237, 

q  =  ao432 1 38.  {i  =  if.) 

Example.    Given  ff  =  io°  23'  46".    Find  q. 

Ans.  \ogg=  7.3 1 563 16. 

Example.    Given  6  =  82*  45'.    Find  g. 

Ans.  log  ^  =  9.37919. 


CHAPTER   XIII. 

NUMERICAL  CALCULATIONS.    K. 

CALCULATION  OF   THE  VALUE   OF  K. 
We  have  already  found  from  eq.  (37),  Chap.  IX, 


(I)  e(o)=    ''''" 


and  from  eq.  (40),  same  chapter, 

0(o)  flK 

But,  eqs.  (38)  and  (27),  Chap.  IX, 

©(A-)  =  I  +  2^  +  2^^  +  2^^  4- 2^- +  .  .  . , 

0(0)  =  I  -  2^  +  2^'  -  2/  +  2^"  -  . . . ; 
whence,  eq.  (2), 

<3)  ^=^(1  +  2^  +  2^^  +  2^'  +  ...)'. 

By  adding  eqs.  (i)  and  (2)  we  get 

0(o)  +  0(Ar)  =  Y/^(i+  1/^); 


whence 


^/Q(o)  +  Q(Ar)V 

2   \       I  +  \'k' 


K=-[    \\\.,4.    '] 


^  n  r2(i+2^-  +  2^''+...)T 
2  L  I  +  |//&'  -1 


+ 

98 


NUMERICAL    CALCULATIONS.      K.  99 

<4)  =  J  (rT^)*('+^?'+^?"+ ■••)'■ 

Example.     Let  ^  =  sin  ^  =  sin  19°  30'.     Required  K. 
First  Method.     By  eq.  (3). 

By  eq.  (5),  Chap.  XII,  we  find  log  q  —  8.6356236.     Apply- 
ing eq.  (3),  using  only  two  terms  of  the  series,  we  have 

\  -\-  2q  —  1.0147662 

log  (i  -|-  2<7)  =  0.0063660 

2  log  (i  +  2<7)  =  0.0127320 

log  -  =  0-1961199 


log  A'=  0.2088519 
K—  1.615101 


Second  Method.     By  eq.  (4). 

Equation  (4)  may  be  written,  neglecting  ^*, 


_  TT/I  -f-  i^cos  b 


2\  2 


whence 


log  cos  e  —  9.9743466, 


logi/cos  ^  =  9-9871733, 


I  -f-  Vcos  ^  =  1-9708973, 


I  +  4/cOS    e  o  o^r 

— ^— =  0.98544865  ; 


and 


log  jr=  0.2088519, 
K—  1.615101, 
the  same  result  as  above. 


ICX)  ELLIPTIC  FUNCTIONS. 

Third  Method.     By  eq.  (7),  Chap.  V. 


e  =  19°  30' 
id  =  9°  45' 
log  tan  iO  =  9.235103 
log  cos  ^6  =  9.993681 

i°f'-'*f  I  =  8.470206 

log  sin  0^  j 

6^,-1°  41' 3i"-i 


6^„=i°4i'3i".i 
i^„  =  0°  50'  45".5 

log  cos  iO,  =  9.999953 


log  cos'  ^6  =  9.987362 
log  cos'  i^„  =  9.999906 

9.987268 
log  -  =  0.196120 


log  K  =  0.208852 

d„„  is  not  calculated,  as  it  is  evident  that  its  cosine  will  be  i. 

Example.     Given  k  =  sin  75°.     Find  K. 

By  eq.  (7),  Chap.  V. 

From  eqs.  (14,),  Chap.  IV,  we  find 


y^  =  sin  0  =  sin  75' 


log  =  9.9849438 


^     tan'i^   -^-^37°  30;  I  ^996.0 

^"        ^  sin  e„      =  sin    36°    4'  i6".47    )  ^^  ^ 


_  jtan'K  =  tan'  18°  2'    8".235  ) 

^»»-1sin^„„     =  sin  6°  5'    9".38    f 

_(tan'i^„„=tan'  3°  2' 34"69    ) 

"  ~  (  sin  ^„3     =  sin  9'  42 ".90    ) 


9.0253880 
7.45 1 1672 


NUMERICAL    CALCULATIONS.      K. 


lOI 


l0£f 


2  loc 


a.  c.  2  loer 


cos  \Q    =  cos  37°  30'  9.8994667  9.7989334  0.2010666 

cos  l^^o  =  cos  18°  2'.i3725  9.9781 184  9.9562368  0.0437632 

cosi^„,  =  cos  3°2'.578i7  9-9993873  9-9987746  0.0012254 

cos  ^(^^^  =  cos    4'.8575  9.9999995  9.9999990  0.00000 10 


re 
2 


0.2460562 
.1961199 


Example.     Given  k  =  sin  45 

Method  of  eq.  (7),  Chap.  V. 

From  eqs.  (14,),  Chap.  IV,  we  have 


log  K  =  0.4421  /  61 

K  =  2.768064    Ans. 
Find  K. 


k      — 


tan' ^^    =  tan' 22°  30' 

sin  6,       =  sin     9°  52'.75683 

tan'  id,   =  tan'  4°  56'.3784i 
sin  6'„„      =  sin  25'.679 

tan^  i6'„„  =  tan'        I2'.3395 
sin  6'„3      =  sin  ©'.05 

a.  c.  log  cos'  ^d  0.0687694 

a.  c.  log  cos'  ^0^  0.0032320 

a.  c.  log  cos'  -l^,,  0.0000060 


log 

9.2344486 


7.8733009 
5-1445523 


1       ^ 
log  — 

^    2 


O.I961199 


log  K  =  0.2681273 

K  =  1.8540747    Ans. 

Example.    Given  6  =  63°  30'.    Find  K. 

Ans.  log  K  =  0.3539686. 

Example.    Given  0  =  34°  30'.     Find  K. 

Ans.  K=  1.72627. 


CHAPTER   XIV. 
NUMERICAL  CALCULATIONS,     u. 
CALCULATION   OF  THE  VALUE   OF  U. 

When  d°  =  sin-'>('  <  45°. 

Example.     Let  0  =  30°,  k  =  sin  45°.     Find  u. 

First  Method.     Eq.   (23),  Chap.  IV,  and  eqs.  (14,),  (14,), 
(143),  Chap.  IV. 

By  equations  (14,), 


-  =  22°  30' ; 


log  tan  ~  =  9.6172243  ; 


log  tan'  -  =  9.2344486  =  log  k,  =  log  sin  6^ ; 


9°  52' 45"  41 ; 


log  tan  -  =  8.9366506  ; 


log  tan^  -"  =  7.8733012  =  log  k,,  =  log  sin  &,, ; 


o°25'4o".7; 


log  tan'  -^°  =  5.144552  =  log  ^.3 . 


NUMERICAL    CALCULATIONS.     U,  IO3 

By  equations  (14,), 

0  =  30° 
log  tan  0  =  9761439 
log  cos  6  =  9.849485 

log  tan  (0,  —  <p)  =  9.610924 

0j  —  0  =  22°  12'  27".56 
(p,  =  52°  12'  27".56 

log  tan  0,  =  0.1 10438 
log  cos  6^„  =  9-993512 

log  tan  (0,  -  0.)  =  0.103949 

0,-0.  =  51°  47'  32".59 

(p,=  i04°o'o".i5 
log  tan  0,  =  0.603228 
log  cos  8,,  =  9  999988 

log  tan  (03  —  0,)  =  0.603216 

0,-0,=  104°  o'  I ".5 

03  =2o8°o'  I ''.65 

Since   ^  =  26°  o' o".04    and     ^' =  26°  o' o".2l, 
4  » 

we  need  not  calculate  0« . 

^-  =  936oo".2i. 


Reducing  this  to  radians,  we  have 
log  |-  =  9.656852. 


€) 


104  ELLIPTIC  FUNCTIONS. 

Substituting  in  eq.  (23),  Chap.  IV,  we  have,  since  cos  ^0,=  i, 

a.  c.  log  cos  6  —  o.\ 505 1 5 

log  cos  ^„  =  9.993512 

log  cos  6^„„  =  9.999988 

0.1440 14 


,  /cos  B^  cos  B 

0.072007  =  log  ^—^^-^ 


log  t'  =  9-656852 


log  11  =  g  728859 

u  =  0.535623,     Ans. 


When  ^=  sin    '^>45°. 


Example.     Given  >^=sin75°,    tan  (f)  =  \/ — r-.  To  find 

F{k,  0). 

First  Method.     Bisected  Amplitudes. 

By  equations  (24)  and  (25),  Chap.  IV,  we  get 

0    =  47°    3'  30^91. 

<t>k  =25°  36'    5".64,  /?    =45°; 

0j  =  13°    6'  3o".98,  A  =  24°  40'  io''.94; 

0j  =    6°  35'  4o".74,  Ao  =  12°  39'  i5"-83  ; 

0A=    3°  18'    8".75,  A3=    6°  22'    8^40; 

0A=    i°39'    /'.43,  A.= 

Substituting  in  equation  (26),  Chap.  IV,  we  have 

F{k,c}>)  =  z2X  i°39'7"-43 
=  52°5i'58".03 
=  0.9226878.    Ans. 


NUMERICAL    CALCULATIONS.       U.  IO5 

Second  Method.     Equation  (29),  Chap.  IV. 
From  equations  (183),  Chap.  IV,  we  have 

log 

y&  =  cos  7  =  COS   15°     O'     0''.00      9.9849438 

^' =  sin  7  =  sin  15°    o'    o".oo     9.4129962 

,,  j  tan' i  ;?  =  tan'  7°  30'  o"-00  )  o  ..oo.o. 
*"  =  }  sin  ,;.  =  sin  0°  59'  IS"-2S  1  '•''''^'' 
k,  =  cos  V.         =  COS    0°  59'  35".25     9.9999348 

|tan'4,.  =  tan.o».9;47;;.62i      3^3^,,^ 

(  Sin  7/„o      =  sin    o     015   .49  ) 
/^2  =  COS  77„„        =  COS    0°    o'  i5".49    o.ooooooo 
k\.  =  ikk'J  1.1493838 

From  equations  (iS,),  Chap.  IV,  we  get 

0  =  47°  3'3o".95; 
200-0  =  45°; 

0„  =  46°  i'45".475; 
0„,  =  46°i'29".4i; 
003  =  46°  i'  29".4i  ; 
45°  +  i03  =  68°o'44".7O5. 

Substituting  these  values  in  eq.  (29),  Chap.  IV,  we  get 

F{k,  0)  =  W  ^  .  j^ .  log  tan  68°  o'  44".705 

=  0.9226877.     Ans. 
Third  Method.     Equation  (23)*,  Chap.  IV. 


106  ELLIPTIC  FUNCTIONS. 

From  equations  (14,),  Chap.  IV,  we  have 

^  =  sin  6^    .        =  sin    75°    o'  o"     log  =  9.984943S 
k'  =  cos  6  =  cos    75°  9.4129962 

,  \  tan=  i  8  =  tan'  37°  30'  )  „^     ^        ' 

(  sin  d^       =  sin    36     4  16  .47  ) 
k/  =  cos  <9„  9.9075648 

,     _    itan=i^„  =  tanM8°   2'8".235]        qo'-:,88(> 
^--  |sin^„„      =sin      6°    5'9".38    f        9.o.5388(> 

/^/  =  cos  6'„„  9-9975452 

^    _   jtan^i^„„  =  tan'    3^    2' 34".69  )         ^..^g.^ 
^--    tsin^„3      =sin  9'42".9of        745II&72 

y^'j  =  COS  6,,  9.9999982 

^^04  =  a  ^ozY  4.3002761 

^/  =  0.0000000 

From  equations  (14,),  Chap.  IV,  we  have 

0  =  47°  3'  3o".94; 
0,=  62°  36'  3".io; 
<P.  =  119°  55'47".67; 
03  =  240°  o'  o".i9; 
0,  =  480°    o'    o''. 

Therefore   the   hmit   of   0,  --  ,  — ,    or     — -  is  30    =  -p. 

24  2"      ^  6 

Substituting  these  values  in  eq.  (23)*,  Chap.  IV,  we  have 

=  0.9226874.     Ans. 
Example.    Given  0  =  30°,  k  =  sin  89°.    Find  u, 
^Method  of  eq.  (28),  Chap.  IV. 


NUMERICAL   CALCULATIONS.     U.  IO7 

From  eqs.  (18,)  we  find 

k^  =  sin  ^,     and     tan'  f  ^,  =  ^  =  sin  ^, 

from  which  we  find  that  /^,  =  i  as  far  as  seven  decimal  places. 
From  eqs,  (iSJ  we  have 

sin  0  =  9.6989700 
k  =  9.9999338 
sin  (20„  —  0)  =  9.6989038 
200  -  0  =  29°  59-69733 
200  =  59°  59'-69733   ' 
45°  +  i0o*=  59°  59-92433 
log  (45° +  i0o)  =  0.2385385     . 

From  eqs.  (18,),  Chap.  IV,  we  have 

k  =  cos  //  =  cos  1°,  iv  =  30'- 

Substituting  in  eq.  (28),  Chap.  IV,  we  have 

a.  c.  log  cos  i  7]       0.0000330 

log  log(45°+i0,)      9.3775585 

a.  clog  J/      0.3622157 


log  F{k,  0)  =  9.7398072 

F{k,  cp)  =  0.549297.     Ans. 

Example.    Given  0  =  79°,    ^  =  0.25882.    Find  u. 

Ans.  u  =  0.39947. 

Example.    Given  0  =  37°,    k'=  0.86603.    Find  u. 

Alls,  u  =  0.68 141. 

*  Since  ki  =  i,  0oo  =  0o,  and  we  need  not  carry  the  calculation  further. 


CHAPTER   XV. 

NUMERICAL   CALCULATIONS.     0. 

Example.    Given  u  =  1.368407,  0  =  38°.    Find  (p. 

First  MetJiod.  From  eqs.  (46)  and  (41)*,  Chap.  IX,  we  have 

u  =  x(d\K\ 

From  equations  (5),  Chap.   XII,  and  (38),  Chap.  IX,  we 
have 

log  ^  =  8.4734187 

log  B\K)  =  0.0501955 

log  u  =  O.I 362 1 53 


log  X  =  0.0860198 
;tr  =  69°  5o'46".i2 

From  equations  (23)  and  (24),  Chap.  IX,  we  get 

log  9,{x)  =  9.9798368 
log  &{x)  =  0.0192687 


9.9605681 
log  Vk'  =  9.9482661 


But 


log  ^0  ==  9.9088342  =  log  sin  A 

k'  sin'  0=1-  JV, 
^  sin  0  =^  cos  A  ; 

108 


whence 


NUMERICAL    CALCULATIONS,      (p.  IO9 

log  COS  A  z=  9.7675483 
log  k  =  9.7893420 


log  sin  0  =  9.9782063 
0  =  72°.     Am. 
Second  Method.   From  eq.  (1),  Chap,  VI. 
From  eqs.  (14,)  Chap.  IV,  we  find 
,         (  tan'i^  =  tan'  19°  )  , 

^-  =  \  sin  ^„       =  sin      6°  48'.54569  \  ^^^  =  9-0739438 
cos  d^  9.9969260 

_  j  tan=  i^„  =  tan'    3°  24'.2784    )  n  r.^^r^r. 

^--|sin^„„      =  I2'.i6659}  7.5488952 

cos  B^^  9.9999974 

(  tan'  ^^„.  =  tan'  6'.o8329  / 

^"=Uni:  \    4.49573.6 

cos   ^03  0.0000000 

Substituting  these  values  in  eq.  (i),  Chap.  VI,  we  have 
log  cos  8^     9  9969260 
log  cos  6^„„     9.9999974 

9-9969234 

log  Vcos  ^„  cos  d^^  9.9984617 

a.  c.  log   "     "  0.0015383 

log  7^   .1362153 

log  i^cos  8  9.9482660 

log  2'      .9030900* 

a.  c.  log  Vcos  6*,  cos  B^^  0.0015383 


0.9891096 
2.2418773 


log  03*  2.7472323 
03  558°  46'.  140 


n  is  taken  equal  to  3,  because  cosos  =  I. 


no  ELLIPTIC  FUNCTIONS. 

Whence,  by  equations  (i)*  of  Chap.  VI,  we  get 

^03     log  =  4.4957316 
sin  03  9-5075232„ 


sin  (20, -03)        4  0032548„ 

20,  -  03  =^  —  o'. 00346 

05  =  279°  23'.o6827 

k,,    log  =:   7.5488952 

sin  0,       9.994 1484„ 
sin  (20,  -  0,)       7.5430436„ 

20,  —  0.J  =  —  I2'.0039 

<?^.  =  139°  35'-532i 

'^o  log  =  9-0739438 
sin  0,       9.8 1 17249 

sin  (20  —  0,)      8.8856687 
20  —  0,  =    4°  24'.467 
0  =  71°  59'.9999 
=  72°.     Ails. 

Example.  Given  u  =  2.41569,    d  —  80°.    Find  0. 

^«.y.  0  =  82°. 
Example.  Given  u  =  1.62530,     J^  =  ^.     Find  0. 

^/W.     0  =   87°. 


CHAPTER   XVI. 

NUMERICAL   CALCULATIONS.     E{k,  0). 

First  Method.  By  Chap.  X,  eqs.  (15),  (16),  and  (17). 
Example.     Given  k  —  0.9327,  0  =  80°.     Find  E{Ji,  (p). 
By  eq.  (15),  Chap.  X, 


0       =80°: 

Y    =  67°  44'.    ; 

0i  =  50°  43'-6, 

y^  =  46°  4o'.4 ; 

0i  =  27°  48'.5, 

n  =  26°    o'.i  ; 

0.   ==  14°  i6'.7, 

n  =13°  24'.o; 

<P.k=    f  1 1 '-3, 

;^,^=    6°45'.2; 

<5^,>,  =    3°  36'.o, 

log 

sin  r^x^  =  8.77094 ; 

0jj  1=  0.062831. 

0^^  <  0.000000 1. 

Whence,  by  eq.  (17), 


^{^,  0^0  =  0.062794 


sin  0      s 

2  sin  0j    s 

4  sin  0j    s 

8  sin  04    s 

16  sin  0^  s 


n'  ;^j  =  0.521 16 
n'xj  =0.29757 
n"  ^j  =  0.10023 
n"  ^^  =  0.02728 
n'  r,^,  =  0.00697 


0.95321 


Hence,  by  eq.  (16), 


-S(>^,  0)  =  32^(/^,  0a)  -  0.95321 

=  2.0094  —  0.9532  =  1.0562. 


m 


112  ELLIPTIC  FUNCTIONS. 

Second  Method.  By  Chap.  X,  eq.  (26). 

Example.     Given   k  =  sin  75°,     tan  (p  =  \  / — -.      Find 

V   1/3 

B{k,  cf>). 

From  eqs.  (Hi),  Chap.  IV,  we  have 

k    —  sin  d  —  sin  75°  o'  o"  log  =  9.9849438 

k'  —  cos  B  =  cos  75°  9.4129962 

I  tan'  id    =  tan'  37°  30'  )  ^     ^ 

^^  =  \  sin  6^      =  sin    36°    4'  i6'\  4?  \        97699610 

k/  =  cos  6^  9.9075648 

(  tan'  i^„  =  tan'  18°    2'    8".235  ) 

^0.  =  -^    •     zj  •        ^o     ./      //    o    >        9.0253880 

"'        ( sin  (^^o     =  sm      6°    5'    9.38    |         ^      ■''^ 

k,'  =  cos  ^„„  9.9975452 

_    (tan'K„  =  tan'    3°    2' 34''-69    ]         ,^,,,672 
^"'~  Isin^,.     =sin  9^42^90    J         74511672 

z^,'  =  cos  ^,3  9.9999982 

>^04   =    (i^os)'  4.3002761 

/^/  =  0.0000000 

From  eqs.  (14,),  Chap.  IV,  we  have 

0  =  47°  3'  3o".94 ; 
0,=  62°  36'  3".  10; 
0,=  119°  55' 47". 67; 
03  ==  240°    o'    o".i9. 


NUMERICAL    CALCULATIONS.      E{k,  0). 

Applying  eq.  (26),  Chap.  X,  we  have 
k''     log  =  9.9698876 


113 


a.  c.  2 


9.6989700 


a.  c.  2 

9.6688576 
9.7699610 
9.6989700 

.4665064 

a.  c.  2 

9.1377886 
9.0253880 
9.6989700 

.1373373 

a.  c.  2 

7.8621466 
7.45 1 1672 
9.6989700 

.0072802 

5.0132838 

.0000103 

I  —  .61 1 1342  =  0.3888658. 


.6111342 


From  eq.  (23)*  Chap.  IV,  we  find  F{k,  (f>)  =  0.9226874. 
Hence 


2 


—  0.3588016 
sin  0,  =       0.3290186 
sin  0,  =       0.0522872 


'—^^  Sin  03  =  -  0.0013888 


k\fk. 


16 


sin  0^  =       0.00000 10 


0.3799180 


114  •       ELLIPTIC  FUNCTIONS. 

Whence 

E{k,  0)  =  0.3588016  +  0.3799180  =  0.7387196.     Ans. 

Example.     Given  k  =  sin  75°.     Find  E\k,  -j. 
From  Example  2,  Chap.  XIII,  we  find 

log/''(^,  ^)  =0.4421761 
log  0.3888658  =  1.5897998 


log£(/^,  j)  =0.0319759 

E\k,  -j  =  1.076405.       Ans. 


Example.     Given  k  =  sin  30°,  0  =  81°.     Find  E{k,  0). 

Ans.  E{k,  0)  =  1. 33 1 24. 
Example.    Find  £(sin  80°,  55°).  Ans.  0.82417. 

Example.    Find  ^(sin  27°,  ^.  Ans.  1.48642. 

Example.    Find  £(sin  19°,  27°).  Ans.  0.46946. 


CHAPTER   XVII. 

APPLICATIONS. 

RECTIFICATION   OF  THE   LEMNISCATE. 


The  polar  equation  of  the  Lemniscate  is  r  =  a  l^cos  28, 
referred  to  the  centre  as  the  origin.     From  this  we  get 

dr  a  sin  2B 


^^  Vcos  26  ' 

whence  the  length  of  the  arc  measured  from  the  vertex  to  any 
point  whose  co-ordinates  are  r  and  0 


=/!G^)'+^f-=«/!^+cos..p 


^ 


=  a 


J    4/cos  28        J    Vi  —  2  sin'  6 
Let  cos  26  =^  cos'  0,  whence 
.dd 


/~-dd>      r  ^ 


sin  0  d<p 


—   cos*  0 
COS  0 


d(p 


r*       d(p a_  n c 

^  ^J^      Vi  +  cos"  0  ~  VjJ^      VT^^i  sin"  0 
-   V2      V2  '^^^ 


115 


Il6  ELLIPTIC  FUNCTIONS. 


Since  r  =  a  -/cos  2^  =  «  cos  (p,  the  angle  0  can  be  easily 
constructed  by  describing  upon  the  axis  a  of  the  Lemniscate  a 
semicircle,  and  then  revolving  the  radius  vector  until  it  cuts 
this  semicircle.  In  the  right-angled  triangle  of  which  this  is  one 
side,  and  the  axis  the  hypotenuse,  (f>  is  evidently  the  angle  be- 
tween the  axis  and  the  revolved  position  of  the  radius  vector. 

RECTIFICATION   OF  THE   ELLIPSE. 

2  3 

Since   the  equation  of  the  ellipse  is   -y+t5=  i,  we  can 

assume  x  ^  a  sin  <{),  y  =1  b  cos  0,  so  that  <f)  is  the  complement 
of  the  eccentric  angle.     Hence 

s  =C  Vdx''  +  d/  =  «y^0  Vi  —  e'  sin"  0 

=  aE{e,  0), 

in  which  e,  the  eccentricity  of  the  ellipse,  is  the  modulus  of  the 
Elliptic  Integral. 

The  length  of  the  Elliptic  Quadrant  is 


s^  =  aE{e,^). 


x'      y 

Example.     The  equation  of  an  ellipse  is -^-^ — [--^^i; 

required  the  length  of  an  arc  whose  abscissas  are  1.061162  and 
4.iocX)00  :  of  the  quadrantal  arc.  Ans.  5.18912;  6.36189. 

rectification  of  the  hyperbola. 

On  the  curve  of  the  hyperbola,  construct  a  straight  line 
perpendicular  to  the  axis  x,  and  at  a  distance  from  the  centre 
equal   to  the  projection   of  b,  the  transverse  axis,  upon   the 

asymptote,  i.e.  equal  to  ■  _-      Join  the  projection  of  the 

y  c^  -\-  b'' 


A  P  PLICA  TIONS.  1 1 7 

given  point  of  the  hyperbola  on  this  line  with  the  centre.  The 
angle  which  this  joining  line  makes  with  the  axis  of  x  we  will 
call  0.  \{  y  is  the  ordinate  of  the  point  on  the  hyperbola,  then 
evidently 

b""  tan  0 
7  = 


and 


a         /         a"  sin*  0  a         /         i     .  „ 


whence 

"^  ^0 


j=    /  |/«^'-f^y  =  - 


V  I 5  si 

dcf) 


cos"  0  r    I 5  sin*  0 


~  (^  J,    cos'  0  4/1  —  /Plin'  0  • 
But 


I  -i^ 


4tan  0  1/1  -  >^'  sin*  0)  =  ^0  ^/j  _  ,^*  sin'  0+^0~^7j^^7^r^^^ 

^0. 


cos'  0  V^i  —  ^'  sin'  0 
Consequently 

b^    /•* ^0 

'c  J^     cos'  0  1/1  -  /fe'  sin'  0 


J  = 


=  -  /\/^,  0)  -  cE{k,  0)  +  ^  tan  0/^(y^,  0) 


=  :^  Aj  '  '^^j  -  ^^^G'  ^)  +  ^^  ^^"  '^  ^l?  *?^)- 


Il8  ELLIPTIC  FUNCTIONS. 

Example,     Find  the  length  of  the  arc  of  the  hyperbola 
I  from  the  vertex  to  the  point  whose  ordinate 


20.25       400 

40 

is tan  15°.  Ans.  5.231 184. 

2.05  ^  J    J       t 

Example.     Find  the  length  of  the  arc  of  the  hyperbola 

77—  =  100  from  the  vertex  to  the  point  whose  ordinate 

144      81  ^ 

is  0.6.  Ans.  0.6582. 


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